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This article is cited in 6 scientific papers (total in 6 papers) On solvability of semilinear second-order elliptic equations on closed manifolds D. V. Tunitskyab a V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow b Lomonosov Moscow State University
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     IntroductionEvolutionary equations of the form where
$$
\begin{equation}
Lu=-\sum_{l,m=1}^n\frac{\partial}{\partial x^l} \biggl(a^{l,m}(x)\,\frac{\partial u}{\partial x^m}\biggr)+ \sum_{l=1}^n b^l(x)\,\frac{\partial u}{\partial x^l}
\end{equation}
\tag{0.2}
$$
is a second-order elliptic linear differential operator, proved widely useful in mathematical modeling of various reaction–diffusion processes, and, in particular, for a description of concentration and propagation of substances, as wells as for balance and development problems of biological populations and genetic features. This topic of great value has been extensively studied. The first advances in this field were made in the famous papers by Kolmogorov, Petrovskii, and Piskunov [1] and Fisher [2], which were concerned with the homogeneous case of equation (0.1), where $Lu=-\sum_{l=1}^n(\partial^2 u/(\partial x^l)^2)$ and $f(x,u)=f(u)$. We also mention the paper [3], which provides a detailed historical background and comprehensive references on this topic. In the case where the solution $u=u(t,x)$ of equation (0.1), (0.2) has the form the left-hand side of this equation reads as
$$
\begin{equation}
\frac{\partial u}{\partial t}+Lu= e^{-\lambda t}\bigl(-\lambda u(x)+Lu(x)\bigr).
\end{equation}
\tag{0.3}
$$
In particular, in the stationary case (with $\lambda=0$), the solution is time-independent and $u=u(x)$ satisfies the semilinear elliptic equation In the present paper, we will consider existence, regularity, and uniqueness of solutions for analogues of equation (0.4) on arbitrary closed manifolds $X$ of finite dimension $n$. Among related papers, we again mention the paper [3], which deals with equation (0.4) with operator $L$ with periodic coefficients. This case, which is reduced to an equation on a manifold diffeomorphic to the $n$-dimensional torus $\mathbb{T}^n$, has great applied value. Of special importance here are analogues of equation (0.4) on other closed manifolds, and, in particular, on those diffeomorphic the $n$-dimensional sphere $\mathbb{S}^n$. In this regard, it is worth pointing out that, according to the generalized Poincaré conjecture, the sphere $\mathbb{S}^n$ is homeomorphic to each closed manifold homotopically equivalent to $\mathbb{S}^n$. The greatest challenge in this conjecture is in the dimension $n=3$. In this case, the proof proposed by G. Ya. Perelman is based on the study of Ricci flows on closed three-dimensional manifolds, see [4], [5]. In essence, these flows are solutions of the corresponding nonlinear systems of partial differential equations. So, the study of solutions of nonlinear partial differential equations on closed manifolds is important and relevant both from applied and mathematical points of view. It is worth pointing out that, in many applied problems, the right-hand sides of equations (0.1) and (0.4) may involve terms which are not smooth or even not continuous. So, it is desirable to be able to select a class of admissible solutions capable of constructing a satisfactory theory of solvability of such equations under minimal requirements on the regularity of their coefficients. In the present paper, we consider the weak solutions as such a class. In this class, it proves possible to investigate solvability of analogues of inhomogeneous equation (0.4) on closed manifolds under fairly light requirements on the regularity of their coefficients. In particular, some coefficients of these equations may be not only discontinuous but also be distributions. It is natural that an increase in regularity of the coefficients results in a corresponding increase in regularity of the solutions. § 1. Function spaces of tensor fieldsBy $X$ we denote an $n$-dimensional smooth closed Riemannian manifold, that is, a connected compact manifold without boundary equipped with the metric $g\colon TX\times TX\to \mathbb{R}$. The metrics induced by $g$ on tensor bundles $(TX)^{\otimes^m}\otimes (T^* X)^{\otimes^l}$, $m,l=0,1,2,\dots$, of this manifold will be denoted by the same letter $g$; further, by $(TX)^{\otimes^0}\otimes(T^* X)^{\otimes^0}$ we will denote the trivial bundle $X \times \mathbb{R}$ with $g(r,t)=rt$ for $r,t \in \mathbb{R}$. The metric $g$ induces on the manifold $X$ the measure $V=V_g$ and the Levi-Civita connection with the covariant differentiation operator $\nabla=\nabla_g$ uniquely defined by this connection. 1.1. Sobolev spacesFrom the above metric $g$ and the measure $V$, for $m,l=0,1,2,\dots$, we define the following norms of tensor fields:
$$
\begin{equation*}
\begin{aligned} \, \|s\|_{L^p((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})}&= \langle g^{p/2}(s,s),1\rangle^{1/p},\qquad p \geqslant 0, \\ \|s\|_{L^\infty((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})}&= \operatorname*{ess\,sup}_{x\in X}\bigl|\sqrt{g(s,s)}\,(x)\bigr|; \end{aligned}
\end{equation*}
\notag
$$
here, for functions $u$ and $v$,
$$
\begin{equation*}
\langle u,v\rangle=\int_X u(x)v(x)\,dV,\qquad \operatorname*{ess\,sup}_{x \in X} u(x)= \inf_{\substack{S \subseteq X \\ V(S)=0}}\, \sup_{x \in X\setminus S} u(x).
\end{equation*}
\notag
$$
We also consider the Banach spaces
$$
\begin{equation*}
\begin{aligned} \, L^p\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr) &=\bigl\{s\colon X\ni x \mapsto s(x) \in (T_x X)^{\otimes^m}\otimes (T_x^* X)^{\otimes^l}\bigm| \\ &\qquad\|s\|_{L^p((TX)^{\otimes^m}\otimes (T^* X)^{\otimes^l})}<+\infty\bigr\}. \end{aligned}
\end{equation*}
\notag
$$
The spaces $L^2\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr)$ are Hilbert spaces with the inner products $\langle s_1,s_2\rangle_{L^2((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})}= \langle g(s_1,s_2),1\rangle$. We also set $L^p(X)=L^p\bigl((TX)^{\otimes^0}\otimes(T^* X)^{\otimes^0}\bigr)$, so that $\langle u,v\rangle_{L^2(X)}=\langle u,v\rangle$. Using the multiple covariant differentiation
$$
\begin{equation*}
\nabla^j\colon C^\infty\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr) \ni s \mapsto \nabla^j s \in C^\infty\bigl((TX)^{\otimes^m}\otimes (T^* X)^{\otimes^{l+j}}\bigr)
\end{equation*}
\notag
$$
we define, for $m,l=0,1,2,\dots$, the norms
$$
\begin{equation*}
\begin{aligned} \, &\|u\|_{W^{k,p}((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})} \\ &\qquad=\biggl(\sum_{j=0}^k\|\nabla^j s\|_{L^p((TX)^{\otimes^m} \otimes (T^* X)^{\otimes^{l+j}})}^p\biggr)^{1/p},\qquad p \geqslant 1, \\ &\|u\|_{W^{k,\infty}((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})} \\ &\qquad=\sum_{j=0}^k\|\nabla^j s\|_{L^\infty((TX)^{\otimes^m} \otimes (T^* X)^{\otimes^{l+j}})}, \qquad k=0,1,2,\dots, \end{aligned}
\end{equation*}
\notag
$$
and the Sobolev spaces
$$
\begin{equation*}
\begin{aligned} \, W^{k,p}\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr) &=\bigl\{s\colon X \ni x \mapsto s(x) \in (T_x X)^{\otimes^m}\otimes (T_x^* X)^{\otimes^l}\bigm| \\ &\qquad \|s\|_{W^{k,p}((TX)^{\otimes^m}\otimes (T^* X)^{\otimes^l})}<+\infty\bigr\}. \end{aligned}
\end{equation*}
\notag
$$
These spaces are completions of the space of infinitely differentiable tensor fields $C^\infty\bigl((TX)^{\otimes^m}\otimes (T^* X)^{\otimes^l}\bigr)$ with respect to the above norms (see, for example, [6], § 10.2.4, and [7], § 7.6). The spaces $W^{k,2}\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr)$ are Hilbert spaces with the inner products
$$
\begin{equation*}
\langle s_1,s_2\rangle_{W^{k,2}((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})}= \sum_{j=0}^k\langle \nabla^j s_1,\nabla^j s_2\rangle_{L^2((TX)^{\otimes^m} \otimes(T^* X)^{\otimes^{l+j}})}.
\end{equation*}
\notag
$$
It is clear that $W^{0,p}\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr)= L^p\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr)$. We define
$$
\begin{equation*}
W^{k,p}(X)=W^{k,p}\bigl((TX)^{\otimes^0}\otimes(T^* X)^{\otimes^0}\bigr).
\end{equation*}
\notag
$$
1.2. Hölder spacesSince the manifold $X$ is compact, the metric $g$ is complete, and hence by the Hopf–Rinow theorem, any two points $x_1,x_2\in X$ can be connected by a minimizing geodesic $\gamma_{x_1,x_2}$ of length $d(x_1,x_2)$. Given each tensor field $s$ on $X$, put
$$
\begin{equation*}
[s]_\alpha=\sup_{x_1,x_2 \in X}\frac{\sqrt{g\bigl(s(\tau_{x_2,x_1}(x_1))- s(x_2),s(\tau_{x_2,x_1}(x_1))-s(x_2)\bigr)}}{d^\alpha(x_1,x_2)},
\end{equation*}
\notag
$$
where $0<\alpha \leqslant 1$ and $\tau_{x_2,x_1}\colon (T_{x_1}X)^{\otimes^m} \otimes (T_{x_1}^* X)^{\otimes^l} \to (T_{x_2}X)^{\otimes^m}\otimes (T_{x_2}^* X)^{\otimes^l}$ is the translation along the curve $\gamma_{x_1,x_2}$ from $x_1$ to $x_2$. Consider the norms
$$
\begin{equation*}
\begin{gathered} \, \|s\|_{C^{k,\alpha}((TX)^{\otimes^m}\otimes (T^* X)^{\otimes^l})}= \|s\|_{C^k((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})}+ [\nabla^k s]_\alpha, \\ k=0,1,2,\dots, \qquad 0<\alpha \leqslant 1, \end{gathered}
\end{equation*}
\notag
$$
where $\|s\|_{C^k((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})}= \|s\|_{W^{k,\infty}((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})}$ is the norm in the space of $k$-smooth tensor fields $C^k\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr)$, $k=0,1,2,\dots$. We also consider the Hölder spaces
$$
\begin{equation*}
\begin{aligned} \, &C^{k,\alpha}\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr) \\ &\qquad=\bigl\{s \in C^k\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr) \bigm| \|s\|_{C^{k,\alpha}((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l})} <+\infty\bigr\}, \end{aligned}
\end{equation*}
\notag
$$
see, for example, [6], § 10.2.4, and [7], § 4.1. It is clear that
$$
\begin{equation*}
C^{k,1}\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr)= W^{k+1,\infty}\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr).
\end{equation*}
\notag
$$
We also set $C^{k,\alpha}(X)=C^{k,\alpha}\bigl((TX)^{\otimes^0}\otimes (T^* X)^{\otimes^0}\bigr)$.
§ 2. The main results2.1. Statement of the problemIn what follows, almost everywhere will be used if some property holds almost everywhere with respect to the measure $V= V_g$. Suppose that, in addition to $g$, the Riemannian manifold $X$ is equipped with another metric $a$. Suppose that this metric is measurable, and there exist positive numbers $a_0$ and $a_1$ such that
$$
\begin{equation}
a_0 g(\eta,\eta)\leqslant a(\eta,\eta) \leqslant a_1 g(\eta,\eta)
\end{equation}
\tag{2.1}
$$
almost everywhere for all $\eta \in T^* X$. Consider the operators $d_a^*$ and $d_g^*$ formally adjoint with the exterior differentiation operator $d$ with respect to the metrics $a$ and $g$, respectively (see [8], Ch. VIII, § 1). In particular, $\langle a(du,v),1\rangle=\langle a(u,d_a^* v),1\rangle$ for all differential $k$-forms $u$ and $(k+1)$-forms $v$, $k=0,1,\dots,n-1$, and if the manifold $X$ is orientable, then $d_a^*=*d*$, where $* = *_a$ is the Hodge operator induced by the metric $a$. Given a function $u\in C^\infty(X)$, we define the linear second-order differential operator where $\Delta=\Delta_a=d_a^*\circ d$ is the geometric Laplacian (the Laplace–de Rahm operator; see [9], Ch. IV, § 5), and $b$ and $c$ are measurable bounded (with respect to the metric $g$) vector field and linear differential form on $X$. In this context, condition (2.1) means that operator (2.2) is uniformly elliptic on the manifold $X$. Consider the measurable function
$$
\begin{equation}
f\colon X \times \mathbb{R} \ni (x,r) \mapsto f(x,r)\in \mathbb{R},
\end{equation}
\tag{2.3}
$$
which is locally Lipschitz continuous almost everywhere in $u$, that is, for each $r \in \mathbb{R}$, there exists a positive constant $\mu_0=\mu_0(r)$ such that
$$
\begin{equation}
|f(\,\cdot\,,r_1)-f(\,\cdot\,,r_2)|\leqslant \mu_0(r)|r_1-r_2|
\end{equation}
\tag{2.4}
$$
almost everywhere for $r_1,r_2 \in [-r,r]$. A weak (or generalized) solution of the equation where $h$ is a measurable bounded linear differential form on the manifold $X$, is a function $u \in W^{1,2}(X)$ such that $ f(\,{\cdot}\,,u)\in L^2(X)$ and
$$
\begin{equation}
\mathcal{L}(u,v)=\langle f(\,\cdot\,,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}
\end{equation}
\tag{2.6}
$$
for each function $v \in C^\infty(X)$, where $\mathcal{L}$ is the continuous bilinear form
$$
\begin{equation}
\mathcal{L}\colon W^{1,2}(X) \times C^\infty(X) \ni (u,v)\mapsto \langle a(du,dv),1\rangle+\langle bu,v\rangle+ \langle uc,dv\rangle_{L^2(T^* X)} \in \mathbb{R}.
\end{equation}
\tag{2.7}
$$
Note that, under the requirements imposed on the input parameters of equation (2.5), not all its coefficients are, in general, ordinary functions. This is related to the fact that the derivatives of the coefficients of the metric $a$ and of the differential forms $c$ and $h$ may prove to be distributions. For example, if the coefficients of the forms $a$, $c$ and $h$ are characteristic functions of sets with regular boundary, then equation (2.5) will involve the corresponding conditions on the values of both the unknown function $u$ and its first derivatives on the boundaries of these sets (see [10], § 6.1). Remark 1. By condition (2.7), $\mathcal{L}(r,v)=r\langle c,dv\rangle_{L^2(T^* X)}$ for each $r \in \mathbb{R}$, and hence, by definition (2.6), the constant function $u=r$ is a weak solution of equation (2.5) if and only if $\theta(r,v)=0$ for any function $v \in C^\infty(X)$, where Remark 2. For a fixed $r$, the functional $\theta(r,\,\cdot\,)$ is a linear form on $C^\infty(X)$. The space $C^\infty(X)$ is dense in $W^{1,2}(X)$, and hence both the linear form $\theta(r,\,\cdot\,)$ (see (2.8)) and the bilinear form $\mathcal{L}$ (see (2.7)) can be uniquely extended by continuity to functions $v \in W^{1,2}(X)$. Moreover, if either of $\theta(r,v)=0$ or (2.6) holds for all functions $v \in C^\infty(X)$, then it also holds for all functions $v \in W^{1,2}(X)$. It is clear that the fulfillment of any of these equalities for all $v \in W^{1,2}(X)$ is equivalent to that for only nonnegative $v \in W^{1,2}(X)$. It is easily seen that, in general, an equation of the form (2.5) may fail to have a weak solution, and, moreover, such a solution (if exists) may be not unique. As simple examples here, we consider the equations $\Delta u=1$ and $\Delta u=0$. The equation $\Delta u=1$ has no weak solutions, because in this case, for $v=1$, in equation (2.6) the left-hand side $\mathcal{L}(u,1)=\langle a(du,d1),1\rangle$ is zero by definition (2.7), while the right-hand side $\langle 1,1\rangle$ is positive; any harmonic function is a solution of the equation $\Delta u=0$. Let us study conditions for existence and uniqueness of a weak solution. 2.2. Existence and regularity of the solutionsA function $u \in W^{1,2}(X)$ is a weak or generalized subsolution (supersolution) of equation (2.5) if $f(\,{\cdot}\,,u)\in L^2(X)$ and if the inequality
$$
\begin{equation*}
Lu \leqslant f+d_g^* h\qquad (Lu \geqslant f+d_g^* h)
\end{equation*}
\notag
$$
is satisfied in the weak sense, that is, for any nonnegative function $v \in C^\infty(X)$,
$$
\begin{equation}
\mathcal{L}(u,v) \leqslant \langle f(\,\cdot\,,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}\quad \bigl(\mathcal{L}(u,v)\geqslant \langle f(\,\cdot\,,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}\bigr).
\end{equation}
\tag{2.9}
$$
A weak subsolution (supersolution) of equation (2.5), which is not its weak solution will be called a strong subsolution (supersolution). In what follows, if not otherwise stated, all subsolutions, solutions, and supersolutions are assumed to be weak, and so for brevity, the attribute “weak” will be omitted. It is clear that each solution $u$ of equation (2.5) is simultaneously its subsolution and supersolution. Conversely, if a function $u$ is simultaneously a subsolution and a supersolution of equation (2.5), then by the concluding argument in Remark 2, it is a solution of equation (2.5). Remark 3. Similarly to Remark 1, a constant function $u=r$, $r \in \mathbb{R}$, is a subsolution (supersolution) of equation (2.5) if and only if $\theta(r,v) \leqslant 0$ ($\theta(r,v) \geqslant 0$) for each nonnegative function $v \in C^\infty(X)$. Moreover, by Remark 2, a subsolution (supersolution) $u$ of equation (2.5) is strong if and only if there exists a nonnegative function $v \in W^{1,2}(X)$ such that Theorem 1 (existence of the solution). Let a metric $a \in L^\infty\bigl((T^* X)^{\otimes^2}\bigr)$ satisfy estimate (2.1), let a vector field $b \in L^\infty(TX)$, let differential forms $c,h \in L^\infty(T^* X)$, let there exist $\mu \in \mathbb{R}$ such that $ \langle c,dv\rangle_{L^2(T^* X)} \geqslant -\langle\mu,v\rangle$ for nonnegative $v \in C^\infty(X)$, and let a function $f \in L_{\rm loc}^\infty(X \times \mathbb{R})$ satisfy the Lipschitz condition (2.4) almost everywhere. Assume that there exist a subsolution $w_0 \in L^\infty(X)$ and a supersolution $w_1 \in L^\infty(X)$ of equation (2.5) satisfying $w_0 \leqslant w_1$ almost everywhere. Then there exist solutions $u_0$ and $u_1$ of equation (2.5) satisfying the estimate almost everywhere; moreover, $u_0$ is minimal and $u_1$ is maximal in the sense that, for each solution $u$ of equation (2.5) satisfying the inequality almost everywhere, the estimate holds almost everywhere. For a proof, see § 3. Suppose that a metric $a \in C^{0,1}\bigl((T^* X)^{\otimes^2}\bigr)$ and linear differential forms $c,h \in C^{0,1}(T^* X)$. By the Rademacher theorem, their coefficients are differentiable almost everywhere. By a strong solution of equation (2.5) we will mean a function $u \in W^{2,p}(X)$, $p \geqslant 1$ or $p=\infty$, satisfying equality (2.5) almost everywhere. Theorem 2 (regularity of the solution). Let the coefficients and the right-hand side of equation (2.5) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) There exists $0<\alpha<1$ such that any $L^\infty(X)$-solution of equation (2.5) lies in $C^{0,\alpha}(X)$. (b) If, for some $0<\alpha<1$, a metric $a \in C^{0,\alpha}\bigl((T^* X)^{\otimes^2}\bigr)$ and linear differential forms $c,h \in C^{0,\alpha}(T^* X)$, then each $L^\infty(X)$-solution of equation (2.5) lies in $C^{1,\alpha}(X)$. (c) If metric $a \in C^{0,1}\bigl((T^* X)^{\otimes^2}\bigr)$ and linear differential forms $c,h\in C^{0,1}(T^* X)$, then each $L^\infty (X)$-solution of equation (2.5) lies in $W^{2,2}(X) \cap C^{1,\alpha}(X)$ for each $0<\alpha<1$, and is a strong solution of this equation. Proof. Under the constraints imposed on the differential operator $L$ (see (2.2)) and the function $f$ (see (2.3)), assertions (a), (b) and (c) follow from well-known properties of solutions of equation (2.5) (see, for example, [7], § 8.9, [7], § 8.11, and [7], § 8.3, respectively).
It is natural, that a further increase in regularity of the coefficients of equation (2.5) results in a corresponding increase of the regularity of its solutions (see, for example, [7], § 8.3). 2.3. Uniqueness of the solutionIn general, the imposed conditions on the coefficients of the operator $L$ (see (2.2)), the function $f$ (see (2.3)), and the linear differential form $h$ in Theorem 1, are insufficient for uniqueness of the solution of equation (2.5). Indeed, it can be shown that a solution may be non-unique even in the case of the homogeneous equation $L=0$ (see, for example, [7], § 8.3). Let us now discuss some questions related to uniqueness of the solution. A function $f$ (see (2.3)) is called strictly right (left) concave at a point $r_0 \in \mathbb{R}$ if
$$
\begin{equation}
(1-\alpha)f(\,\cdot\,,r_0)+\alpha f(\,\cdot\,,r)< f(\,\cdot\,,(1-\alpha)r_0+\alpha r)
\end{equation}
\tag{2.13}
$$
almost everywhere for $r>r_0$ ($r<r_0$) and $0<\alpha<1$. Similarly, a function $f$ is strictly right (left) convex at a point $r_0\in \mathbb{R}$ if
$$
\begin{equation}
f\bigl(\,\cdot\,,(1-\alpha)r_0+\alpha r\bigr)< (1-\alpha)f(\,\cdot\,,r_0)+\alpha f(\,\cdot\,,r)
\end{equation}
\tag{2.14}
$$
almost everywhere for $r>r_0$ ($r<r_0$) and $0<\alpha<1$. The following result holds. Theorem 3 (uniqueness of the solution). Let the coefficients and the right-hand side of equation (2.5) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let $u_1$ and $u_2$ be solutions, and let a constant $r_0 \in \mathbb{R}$ be a subsolution of equation (2.5) with $r_0<u_1$, $r_0<u_2$ almost everywhere. Assume also that the function $f$ (see (2.3)) is strictly right concave at a point $r_0 \in \mathbb{R}$. Then $u_1=u_2$ almost everywhere. (b) Assume that $u_1$ and $u_2$ are solutions, and a constant $r_0\in \mathbb{R}$ is a supersolution of equation (2.5) with $u_1<r_0$, $u_2<r_0$ almost everywhere. Assume also that the function $f$ (see (2.3)) is strictly left convex at a point $r_0\in \mathbb{R}$. Then $u_1=u_2$ almost everywhere. For a proof, see § 4. A similar result for classical solutions in $\mathbb{R}^n$ in the case where the manifold $X$ is an $n$-dimensional torus $\mathbb{T}^n$ is given in [3], § 3.2. Theorem 4 (constancy of a solution). Let the coefficients and the right-hand side of equation (2.5) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let $r_0 \in \mathbb{R}$, $w \in W^{1,2}(X) \cap L^\infty(X)$, $w \geqslant 0$ almost everywhere, and let $r_0+\alpha w$ be a strong supersolution of equation (2.5) for $0<\alpha \leqslant A$, where $A>0$. Next, let $u$ be a solution of equation (2.5) such that $r_0 \leqslant u \leqslant r_0+Aw$ almost everywhere. Then $u=r_0$ almost everywhere. (b) Let $r_0 \in \mathbb{R}$, $w \in W^{1,2}(X) \cap L^\infty(X)$, $w \leqslant 0$ almost everywhere, and let $r_0+\alpha w$ be a strong subsolution of equation (2.5) for $0<\alpha \leqslant A$, where $A>0$. Next, let $u$ be a solution of equation (2.5) such that $r_0+Aw \leqslant u \leqslant r_0$ almost everywhere. Then $u=r_0$ almost everywhere. For a proof, see § 4. Theorem 5 (alternativity of the solution). Let the coefficients and the right-hand side of equation (2.5) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let a constant $r_0 \in \mathbb{R}$ be a supersolution of equation (2.5), and let $u_1$ and $u_2$ be solutions of equation (2.5) such that $r_0 \leqslant u_1 \leqslant u_2$ almost everywhere. Let the function $f$ (see (2.3)) be strictly right convex at the point $r_0$. Then either $u_1=r_0$, or $u_1=u_2$. (b) Let a constant $r_0 \in \mathbb{R}$ be a subsolution of equation (2.5), and let $u_1$, $u_2$ be solutions of equation (2.5) such that $u_2 \leqslant u_1 \leqslant r_0$ almost everywhere. Let the function $f$ (see (2.3)) be strictly left concave at $r_0$. Then either $u_1=r_0$, or $u_1=u_2$. For a proof, see § 4. 2.4. Eigenvalue problemsIdentity (0.3) from the introduction requires a certain consideration and leads to the following constructions (cf. [3], § 2). Suppose that the function $f$ (see (2.3)) has at $r_0 \in \mathbb{R}$ the right (left) derivative
$$
\begin{equation*}
\begin{gathered} \, \frac{\partial f}{\partial r}(\,\cdot\,,r_0+0)=\lim_{r \to r_0+0} \frac{f(\,\cdot\,,r)-f(\,\cdot\,,r_0)}{r-r_0} \\ \biggl(\frac{\partial f}{\partial r}(\,\cdot\,,r_0-0)= \lim_{r \to r_0-0}\frac{f(\,\cdot\,,r)-f(\,\cdot\,,r_0)}{r-r_0}\biggr) \end{gathered}
\end{equation*}
\notag
$$
almost everywhere. For the differential operator $L$ ((2.2)), we pose the following eigenvalue problem with the potential
$$
\begin{equation}
F(x)=\frac{\partial f}{\partial r}(x,r_0+0)\qquad \biggl(F(x)=\frac{\partial f}{\partial r}(x,r_0-0)\biggr).
\end{equation}
\tag{2.16}
$$
Assume that a function $w \in W^{1,2}(X) \cap L^\infty(X)$ is essentially different from zero, that is, $\|u\|_{L^\infty(X)}>0$. We will say that $w$ is a subeigenfunction (supereigenfunction) of problem (2.15), (2.16) belonging to a value $\lambda \in \mathbb{R}$ if the inequality
$$
\begin{equation}
\langle Lw-Fw,v\rangle \leqslant \lambda\langle w,v\rangle\qquad (\langle Lw-Fw,v\rangle \geqslant \lambda\langle w,v\rangle)
\end{equation}
\tag{2.17}
$$
holds for all nonnegative functions $v \in C^\infty(X)$. It is natural here that if (2.17) are equalities, then $w$ is an eigenfunction $\lambda$ is an eigenvalue. Theorem 6 (existence of the solution). Let the coefficients and the right-hand side of equation (2.5) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let the function $f$ (see (2.3)) have at $r_0 \in \mathbb{R}$ the right derivative from $L^\infty(X)$ almost everywhere, and let $w$ be a subeigenfunction of the corresponding problem (2.15), (2.16) belonging to a negative value $\lambda$ for which $\operatorname*{ess\,inf}_{x \in X}w(x)> 0$. Next, let numbers $r_0$ and $N$, $N>r_0$, be, respectively, a subsolution and supersolution of equation (2.5). Then this equation has a solution $u$ satisfying $r_0<u \leqslant N$ almost everywhere. (b) Let the function $f$ (see (2.3)) have at $r_0 \in \mathbb{R}$ the left derivative from $L^\infty(X)$ almost everywhere, and let $w$ be a supereigenfunction of the corresponding problem (2.15), (2.16) belonging to a negative value $\lambda$ for which $\operatorname*{ess\,sup}_{x \in X}w(x)<0$. Next, let numbers $r_0$ and $N$, $N<r_0$, be, respectively, a supersolution and a subsolution of equation (2.5). Then this equation has a solution $u$ satisfying $N \leqslant u<r_0$ almost everywhere. The proof, which is given in § 5, shows that in the actual fact the solutions $u$ satisfy somewhat stronger estimates. Note that by assertion (a) of Theorem 2 the solution $u$ is continuous, and hence is uniformly separated from $r_0$. From definition (2.13) it readily follows that the function $f$ (see (2.3)) is strictly right (left) concave at a point $r_0 \in \mathbb{R}$ if and only if
$$
\begin{equation}
\frac{f(\,\cdot\,,r_1)-f(\,\cdot\,,r_0)}{r_1-r_0}< \frac{f(\,\cdot\,,r)-f(\,\cdot\,,r_0)}{r-r_0}
\end{equation}
\tag{2.18}
$$
almost everywhere for $r_0<r<r_1$ ($r_1<r<r_0$). Similarly, from definition (2.14) it easily follows that a function $f$ is strictly right (left) convex at a point $r_0 \in \mathbb{R}$ if and only if
$$
\begin{equation}
\frac{f(\,\cdot\,,r_1)-f(\,\cdot\,,r_0)}{r_1-r_0}> \frac{f(\,\cdot\,,r)-f(\,\cdot\,,r_0)}{r-r_0}
\end{equation}
\tag{2.19}
$$
almost everywhere for $r_0<r<r_1$ ($r_1<r<r_0$). These monotonicity properties of the difference relations imply the existence of the corresponding derivatives in the right-hand sides of (2.16), and, as a corollary, imply that problem (2.15) is well posed (for further details, see Lemma 5 in § 5). Now the following result is a consequence of Theorems 6 and 3. Theorem 7 (existence and uniqueness of the solution). Let the coefficients and the right-hand side of equation (2.5) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let the function $f$ be strictly right concave at a point $r_0 \in \mathbb{R}$ and $w$ be a subeigenfunction function of problem (2.15), (2.16) belonging to a negative value $\lambda$ for which $\operatorname*{ess\,inf}_{x\in X}w(x)>0$. Next, let numbers $r_0$ and $N$, $r_0<N$, be a subsolution and a supersolution of equation (2.5). Then this equation has a unique solution $u$ satisfying $r_0<u$ almost everywhere. (b) Let the function $f$ be strictly left convex at a point $r_0 \in \mathbb{R}$ and $w$ be a supereigenfunction function of the corresponding problem (2.15), (2.16) belonging to a negative value $\lambda$ for which $\operatorname*{ess\,sup}_{x \in X}w(x)<0$. Next, let numbers $r_0$ and $N$, $r_0>N$, be, respectively, a supersolution and a subsolution of equation (2.5). Then this equation has a unique solution $u$ satisfying $u<r_0$ almost everywhere. The following result supplements Theorem 7. Theorem 8 (constancy of the solution). Let the coefficients and the right-hand side of equation (2.5) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let the function $f$ be strictly right concave at a point $r_0 \in \mathbb{R}$, and $w$ be a supereigenfunction of problem (2.15), (2.16) belonging to a nonnegative value $\lambda$ for which $\operatorname*{ess\,inf}_{x \in X}w(x)>0$. Next, let a number $r_0$ be a supersolution, and a function $u$ be a solution of equation (2.5) satisfying the inequality $r_0 \leqslant u$ almost everywhere. Then $u=r_0$ almost everywhere. (b) Let the function $f$ be strictly left convex at a point $r_0 \in \mathbb{R}$ and $w$ be a subeigenfunction of problem (2.15), (2.16) belonging to a nonnegative value $\lambda$ for which $\operatorname*{ess\,sup}_{x\in X}w(x)<0$. Next, let a number $r_0$ be a subsolution, and a function $u$ be a solution of equation (2.5) satisfying the inequality $u \leqslant r_0$ almost everywhere. Then $u=r_0$ almost everywhere. For a proof, see § 5. If in the operator $L$ (see (2.2)) we have $b=0$ and $c=0$, then equation (2.5) assumes the form and problem (2.15) now reads as Since $\Delta-F$ is a self-adjoint elliptic operator, its minimal eigenvalue $\lambda_1$ is a real and simple, and the function $w_1$ belonging to the eigenvalue $\lambda_1$ can be chosen positive, see, for example [7], § 8.12. Hence in our setting from Theorems 6, 7 and 8 we have the following results.Corollary 1. Let the coefficients and the right-hand side of equation (2.20) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let the function $f$ (see (2.3)) have at a point $r_0 \in \mathbb{R}$ the right derivative from $L^\infty(X)$ almost everywhere, and let the minimal eigenvalue $\lambda_1$ of problem (2.16), (2.21) be negative. Next, let numbers $r_0$ and $N$, $N>r_0$, be, respectively, a subsolution and a supersolution of equation (2.20). Then this equation has a solution $u$ satisfying $r_0<u \leqslant N$ almost everywhere. (b) Let the function $f$ (see (2.3)) have at a point $r_0 \in \mathbb{R}$ the left derivative from $L^\infty(X)$ almost everywhere, and let the minimal eigenvalue $\lambda_1$ of problem (2.16), (2.21) be negative. Next, let numbers $r_0$ and $N$, $N<r_0$, be, respectively, a supersolution and a subsolution of equation (2.20). Then this equation has a solution $u$ satisfying $N \leqslant u<r_0$ almost everywhere. The proof is a direct application of Theorem 6. Corollary 2. Let the coefficients and the right-hand side of equation (2.20) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let the function $f$ be strictly right concave at a point $r_0 \in \mathbb{R}$ and let the minimal eigenvalue $\lambda_1$ of problem (2.16), (2.21) be negative. Next, let numbers $r_0$ and $N$, $N>r_0$, be, respectively, a subsolution and a supersolution of equation (2.20). Then this equation has a unique solution $u$ satisfying $r_0<u$ almost everywhere. (b) Let the function $f$ be strictly left convex at a point $r_0 \in \mathbb{R}$ and let the minimal eigenvalue $\lambda_1$ of problem (2.16), (2.21) be negative. Next, let numbers $r_0$ and $N$, $N<r_0$, be, respectively, a supersolution and a subsolution of equation (2.20). Then this equation has a unique solution $u$ satisfying $u<r_0$ almost everywhere. The proof is a direct application of Theorem 7. Corollary 3. Let the coefficients and the right-hand side of equation (2.20) satisfy the hypotheses of Theorem 1. Then the following results hold. (a) Let the function $f$ be strictly right concave at a point $r_0 \in \mathbb{R}$ and let the minimal eigenvalue $\lambda_1$ of problem (2.16), (2.21) be nonnegative. Next, let a number $r_0$ be a supersolution, and $u$ be a solution of equation (2.20) satisfying the inequality $r_0 \leqslant u$ almost everywhere. Then $u=r_0$ almost everywhere. (b) Let the function $f$ be strictly left convex at a point $r_0 \in \mathbb{R}$ and let the minimal eigenvalue $\lambda_1$ of problem (2.16), (2.21) be nonnegative. Next, let a number $r_0$ be a subsolution, and $u$ be a solution of equation (2.20) satisfying the inequality $u \leqslant r_0$ almost everywhere. Then $u=r_0$ almost everywhere. The proof is a direct application of Theorem 8. § 3. Existence of the solution3.1. Auxiliary resultThe following auxiliary lemma will be required in the proof of Theorem 1. Lemma 1. Let the coefficients of the differential operator $L$ (see (2.2)) satisfy the hypotheses of Theorem 1. Then the following assertions hold. (a) There exists a positive number $\mu_1$ such that, for $\mu \geqslant \mu_1$ and $u \in W^{1,2}(X)$,
$$
\begin{equation*}
\mathcal{L}(u,u) \geqslant \frac{a_0}{2} \langle du,du\rangle_{L^2(T^* X)}- \mu\langle u,u\rangle.
\end{equation*}
\notag
$$
(b) There exist positive numbers $\mu_2$ and $C=C(\mu_2)$ such that, for $\mu \geqslant \mu_2$ and $u \in W^{1,2}(X)$, the following a priori estimate holds:
$$
\begin{equation*}
\langle u,u\rangle_{W^{1,2}(X)} \leqslant C(\mathcal{L}(u,u)+\mu\langle u,u\rangle).
\end{equation*}
\notag
$$
(c) There exists a positive number $\mu_3$ such that if, for $\mu \geqslant \mu_3$, functions $u_0,u_1 \in W^{1,2}(X)$ scarify the inequality for any nonnegative function $v \in C^\infty(X)$, then $u_0 \leqslant u_1$ almost everywhere. Proof. The proof of assertion (a) is by estimation of the left-hand side of the required inequality using definition (2.7) and the uniform ellipticity condition (2.1) (see, for example, [7], § 8.2). Assertion (b) is a clear corollary to assertion (a). Assertion (c) follows from the strong maximum principle for weak subsolutions (see, for example, [7], § 8.7). Lemma 1 is proved. 3.2. Proof of Theorem 1The function $f$ (see (2.3)) satisfies the Lipschitz condition (2.4) almost everywhere, and hence, for any
$$
\begin{equation*}
\mu \geqslant \mu_0 (\max\{\|w_0\|_{L^\infty(X)},\|w_1\|_{L^\infty(X)}\})
\end{equation*}
\notag
$$
and any functions $v_0$ and $v_1$ satisfying the estimate $w_0 \leqslant v_0 \leqslant v_1 \leqslant w_1$ almost everywhere, the inequality $f(\,\cdot\,,v_0)+\mu v_0 \leqslant f(\,\cdot\,,v_1)+\mu v_1$ holds almost everywhere (cf. [11], Appendix to Ch. IV, § 2). It follows that
$$
\begin{equation}
\langle f(\,\cdot\,,v_0)+\mu v_0,v\rangle+ \langle h,dv\rangle_{L^2(T^* X)} \leqslant \langle f(\,\cdot\,,v_1)+\mu v_1,v\rangle+\langle h,dv\rangle_{L^2(T^*X)},
\end{equation}
\tag{3.1}
$$
for nonnegative functions $v \in C^\infty(X)$. We put
$$
\begin{equation}
c_0=\max\bigl\{\mu_0(\max\{\|w_0\|_{L^\infty(X)},\|w_1\|_{L^\infty(X)}\}), \mu_1,\mu_2,\mu_3\bigr\},
\end{equation}
\tag{3.2}
$$
where $\mu_1$, $\mu_2$ and $\mu_3$ are the positive numbers from assertions (a), (b) and (c) of Lemma 1. It is clear that equation (2.5) is equivalent to the equation We construct by induction two sequences of approximations $\{u_{0,l}\}$ and $\{u_{1,l}\}$: let $u_{j,0}=w_j$ for $j=0,1$, and let $u_{j,l}$, $l=1,2,\dots$, be a solution of the equation
$$
\begin{equation*}
Lu_{j,l}+c_0 u_{j,l}=f(\,\cdot\,,u_{j,l-1})+d_g^* h+c_0 u_{j,l-1},
\end{equation*}
\notag
$$
that is, for each function $v \in C^\infty(X)$,
$$
\begin{equation}
\mathcal{L}(u_{j,l},v)+c_0\langle u_{j,l},v\rangle= \langle f(\,\cdot\,,u_{j,l-1})+c_0 u_{j,l-1},v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}.
\end{equation}
\tag{3.4}
$$
By assertion (a) of Lemma 1, the bounded quadratic form $\mathcal{L}(u,u)+c_0\langle u,u\rangle$ is coercive in the Hilbert space $W^{1,2}(X)$. By the Lax–Milgram theorem, equation (3.4) is uniquely solvable in this space (see, for example, [7], § 5.8). For each solution $u$ satisfying estimate (2.11), from inequality (3.1) and assertion (c) of Lemma 1 we have, almost everywhere,
$$
\begin{equation*}
w_0=u_{0,0} \leqslant u_{0,1} \leqslant u_{0,2} \leqslant\dots\leqslant u \leqslant\dots\leqslant u_{1,2} \leqslant u_{1,1} \leqslant u_{1,0}=w_1.
\end{equation*}
\notag
$$
Since the sequences $\{u_{0,l}\}$ and $\{u_{1,l}\}$ are monotone and bounded, their limits
$$
\begin{equation}
u_0=\lim_{l \to +\infty} u_{0,l},\qquad u_1=\lim_{l \to +\infty}u_{1,l}
\end{equation}
\tag{3.5}
$$
exist almost everywhere. For these limits,
$$
\begin{equation}
w_0=u_{0,0} \leqslant u_{0,1} \leqslant u_{0,2} \leqslant\dots\leqslant u_0 \leqslant u \leqslant u_1 \leqslant\dots\leqslant u_{1,2} \leqslant u_{1,1} \leqslant u_{1,0}=w_1
\end{equation}
\tag{3.6}
$$
almost everywhere. Hence by the Lebesgue dominated convergence theorem, equalities (3.5) hold also in the norm ${\|\cdot\|_{W^{0,2}(X)}}$. Next, by the choice of $c_0$ (see (3.2)) and assertion (b) of Lemma 1, there exists a constant $C$ such that, for $j=0,1$ and all $l,m=1,2,\dots$,
$$
\begin{equation*}
\begin{aligned} \, \langle u_{j,l+m}-u_{j,l},u_{j,l+m}-u_{j,l}\rangle_{W^{1,2}(X)} &\leqslant C\bigl(\mathcal{L}(u_{j,l+m}-u_{j,l},u_{j,l+m}-u_{j,l}) \\ &\qquad +c_0\langle u_{j,l+m}-u_{j,l},u_{j,l+m}-u_{j,l}\rangle\bigr). \end{aligned}
\end{equation*}
\notag
$$
By definition of the form $\mathcal{L}$ (see (2.7)), we have
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{L}(u_{j,l+m}-u_{j,l},u_{j,l+m}-u_{j,l})+ c_0\langle u_{j,l+m}-u_{j,l},u_{j,l+m}-u_{j,l}\rangle \\ &\qquad=\mathcal{L}(u_{j,l+m},u_{j,l+m}-u_{j,l}) +c_0\langle u_{j,l+m},u_{j,l+m}-u_{j,l}\rangle \\ &\qquad\qquad-\mathcal{L}(u_{j,l},u_{j,l+m}-u_{j,l}) -c_0\langle u_{j,l},u_{j,l+m}-u_{j,l}\rangle, \end{aligned}
\end{equation*}
\notag
$$
and hence, by construction (3.4),
$$
\begin{equation*}
\begin{aligned} \, &\langle u_{j,l+m}-u_{j,l},u_{j,l+m}-u_{j,l}\rangle_{W^{1,2}(X)} \\ &\qquad\leqslant C\langle f(\,\cdot\,,u_{j,l+m-1})+c_0 u_{j,l+m-1},u_{j,l+m}-u_{j,l}\rangle \\ &\qquad\qquad-C\bigl\langle f(\,\cdot\,,u_{j,l-1})+ c_0 u_{j,l-1},u_{j,l+m}-u_{j,l}\bigr\rangle. \end{aligned}
\end{equation*}
\notag
$$
Therefore, by the Lebesgue dominated convergence theorem, the above approximating sequences $\{u_{0,l}\}$ and $\{u_{1,l}\}$ are Cauchy sequences with respect to the norm $\|\,{\cdot}\,\|_{W^{1,2}(X)}$. Hence they converge to functions $u_0$ and $u_1$, respectively (see (3.5)), so that, passing to the limit in (3.4), for $j=0,1$ we get
$$
\begin{equation*}
\mathcal{L}(u_j,v)+c_0\langle u_j,v\rangle= \langle f(\,\cdot\,,u_j),v\rangle+ c_0\langle u_j,v\rangle+\langle h,dv\rangle_{L^2(T^* X)}
\end{equation*}
\notag
$$
for $v \in C^\infty(X)$, that is, the functions $u_0$ and $u_1$ are solutions of equation (3.3). Inequalities (2.10) and (2.12) for $u$ follow from estimate (3.6). Theorem 1 is proved.
§ 4. Uniqueness of the solution4.1. Auxiliary resultsThe following lemmas will be required in the proof of the theorem on uniqueness of the solution. Lemma 2. Let the coefficients and the right-hand side of equation (2.5) satisfy the hypotheses of Theorem 1 and let $u,u_0 \in W^{1,2}(X) \cap L^\infty(X)$ be such that $ u_0 \leqslant u$ almost everywhere. Next, let $u_0$, $u$ be a subsolution and a supersolution of equation (2.5) such that $\operatorname*{ess\,sup}_{x \in X}(u_0-u)(x)=0$. Then $u_0=u$ almost everywhere, and, therefore, $u$ and $u_0$ are solutions of equation (2.5). Proof. By the assumption, $u_0$ is a subsolution, and $u$ is a supersolution of equation (2.5). Their difference $w=u_0-u $ is non-positive almost everywhere. By definition (2.9) and from the Lipschitz condition (2.4), we have
$$
\begin{equation*}
\begin{aligned} \, 0 &\geqslant \mathcal{L}(w,v)-\langle f(\,\cdot\,,u_0)- f(\,\cdot\,,u),v\rangle \\ &\geqslant \mathcal{L}(w,v)+ \bigl(\mu_0(\max\{\|u_0\|_{L^\infty(X)},\|u\|_{L^\infty(X)}\})\bigr) \langle w,v\rangle \end{aligned}
\end{equation*}
\notag
$$
for all nonnegative $v \in C^\infty(X)$. Hence the difference $w$ is a subsolution of the equation $Lw+\mu w=0$ for any $\mu \in \mathbb{R}$ such that $\mu \geqslant \mu_0(\max\{\|u_0\|_{L^\infty(X)},\|u\|_{L^\infty(X)}\})$. By the assumption, $\operatorname*{ess\,sup}_{x\in X}w(x)=0$ and since $X$ is compact, it follows that $w=0$ almost everywhere by the strong maximum principle for subsolutions (see, for example, [7], § 8.7). Lemma 2 is proved. Lemma 3. For a functional $\mathcal{L}$ (see (2.7)), any numbers $r,\alpha \in \mathbb{R}$, and any functions $w,v \in W^{1,2}(X)$, Proof. By Remark 2, the bilinear form $\mathcal{L}$ (see (2.7)) and the linear form $\theta(r_0,\,\cdot\,)$ (see (2.8)) in equalities (4.1) and (4.2) are defined correctly. By Remark 1,
$$
\begin{equation*}
\mathcal{L}(r+\alpha w,v)=\mathcal{L}(r,v)+\mathcal{L}(\alpha w,v)= \alpha\mathcal{L}(w,v)+r\langle c,dv\rangle_{L^2(T^* X)},
\end{equation*}
\notag
$$
which proves (4.1). Since $u$ is a solution of equation (2.5), from (4.1) we get (4.2) with $w=u$ and $r=(1-\alpha)r_0$ by definition (2.6) and by construction of $\theta$. Lemma 3 is proved. Lemma 4. Let $u$ be a solution, let a constant $r_0 \in \mathbb{R}$ be a subsolution (a supersolution) of equation (2.5), and let the function $f$ (see (2.3)) be either strictly right (left) convex or concave at a point $r_0 \in \mathbb{R}$ ($r_0<u$ or $r_0>u$ almost everywhere, respectively). Then, for $0<\alpha<1$, the function $r_0+\alpha(u-r_0)$ is a strong subsolution (a supersolution) of equation (2.5). Proof. In the case where a constant $r_0 \in \mathbb{R}$ is a subsolution of equation (2.5), by Remark 3 we have $\theta(r_0,v)\leqslant 0$ for each nonnegative function $v \in C^\infty(X)$. Hence by equality (4.2) from Lemma 3 and the definition of concavity (2.13), for nonnegative functions $v \in C^\infty(X)$, we have
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{L}\bigl(r_0+\alpha(u-r_0),v\bigr)= \alpha\langle f(\,\cdot\,,u),v\rangle+(1-\alpha)\theta(r_0,v)+ (1-\alpha)\langle f(\,\cdot\,,r_0),v\rangle \\ &\qquad+\langle h,dv\rangle_{L^2(T^* X)}\leqslant\bigl\langle f(\,\cdot\,,(1-\alpha)r_0+\alpha u),v\bigr\rangle+\langle h,dv\rangle_{L^2(T^* X)} \end{aligned}
\end{equation*}
\notag
$$
for $0<\alpha<1$, and moreover, since inequality (2.13) is strict, for $v=1$ we have
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{L}(r_0+\alpha(u-r_0),1)=\alpha\langle f(\,\cdot\,,u),1\rangle+ (1-\alpha)\theta(1,v)+(1-\alpha)\langle f(\,\cdot\,,r_0),1\rangle \\ &\qquad\leqslant \langle\alpha f(\,\cdot\,,u)+(1-\alpha)f(\,\cdot\,,r_0),1\rangle <\langle f(\,\cdot\,,(1-\alpha)r_0+\alpha u),1\rangle. \end{aligned}
\end{equation*}
\notag
$$
So, for $0<\alpha<1$, the function $r_0+\alpha(u-r_0)$ is a strong subsolution of equation (2.5) in accordance with definition (2.9) and Remark 3. The case where a constant $r_0 \in \mathbb{R}$ is a supersolution of equation (2.5) is considered similarly. Lemma 4 is proved. 4.2. Proof of Theorem 3By the assumption, the manifold $X$ is compact, and by assertion (a) of Theorem 2, we can assume that the solutions $u_1$ and $u_2$ are continuous. (a) By the assumption, the constant $r_0$ is a subsolution of equation (2.5), and the function $f$ (see (2.3)) is strictly right concave at a point $r_0 \in \mathbb{R}$, and hence the function $r_0+\alpha(u_j - r_0)$ is a strong subsolution of equation (2.5) for $0<\alpha<1$ and $j=1,2$ by Lemma 4. Since the solutions $u_1$ and $u_2$ are continuous, and since $r_0<u_1$, $r_0<u_2$ almost everywhere, it follows that, for $j=1,2$,
$$
\begin{equation*}
\alpha_j=\sup\{\alpha>0 \mid u_{3-j} \geqslant r_0+\alpha(u_j-r_0)\text{ almost everywhere}\}
\end{equation*}
\notag
$$
exists and is a positive number (cf. [3], § 3.2). If $0 < \alpha_j < 1$, then we have $\operatorname*{ess\,sup}_{x \in X} \bigl(r_0+\alpha_j(u_j-r_0)-u_{3-j}\bigr)(x)=0$, and $u_{3-j}=r_0+\alpha_j(u_j-r_0)$ almost everywhere by construction of $\alpha_j$ and by Lemma 2. But this contradicts the fact that the function $r_0+\alpha_j(u_j- r_0)$ is not a solution of equation (2.5). Therefore, $\alpha_j \geqslant 1$, and hence $u_{3-j} \geqslant r_0+\alpha_j(u_j-r_0) \geqslant r_0+1(u_j-r_0)=u_j$ almost everywhere for $j=1,2$. Case (b) is dealt with similarly. Theorem 3 is proved. 4.3. Proof of Theorem 4(a) We have $r_0 \leqslant u \leqslant r_0+Aw$ almost everywhere, and hence
$$
\begin{equation*}
\alpha_0=\inf\{0<\alpha \leqslant A\mid u \leqslant r_0+\alpha w \text{ almost everywhere}\}
\end{equation*}
\notag
$$
exists and $0 \leqslant \alpha_0 \leqslant A$ (cf. [3], § 3.2). Therefore, if $\alpha_0>0$, then we have $\operatorname*{ess\,sup}_{x \in X} (u-r_0-\alpha_0 w)(x)=0$, and since the function $r_0+\alpha w$ is a strong supersolution of equation (2.5) for $0<\alpha \leqslant A$, we have $u=r_0+\alpha_0 w$ almost everywhere by Lemma 2, but this contradicts the fact that $r_0+\alpha_0 w$ is not a solution. Therefore, $\alpha_0=0$ and $u=r_0$ almost everywhere. Case (b) is dealt with similarly. Theorem 4 is proved. 4.4. Proof of Theorem 5The manifold $X$ is compact, and by assertion (a) of Theorem 2, it can be assumed without loss of generality that the solutions $u_1$ and $u_2$ are continuous. (a) By the assumption, $r_0\leqslant u_1\leqslant u_2$ almost everywhere, and hence
$$
\begin{equation*}
\varepsilon=-\operatorname*{ess\,sup}_{x \in X}(u_1-u_2)(x)
\end{equation*}
\notag
$$
is nonnegative. If $\varepsilon=0$, then $u_1=u_2$ almost everywhere by Lemma 2, proving the theorem. If $\varepsilon>0$, then since $u_2$ is continuous and $X$ is compact, the quantity $R=\|u_2-r_0\|_{L^\infty(X)}$ is finite, so that,
$$
\begin{equation*}
0 \leqslant u_1-r_0 \leqslant u_2-r_0-\varepsilon \leqslant u_2-r_0-\varepsilon\frac{u_2-r_0}{R}= \biggl(1-\frac{\varepsilon}{R}\biggr)(u_2-r_0)
\end{equation*}
\notag
$$
almost everywhere, and hence $r_0 \leqslant u_1 \leqslant r_0+(1-\varepsilon/R)(u_2-r_0)$ almost everywhere. Since the constant $r_0$ is a supersolution of equation (2.5) and the function $f$ is strictly right convex at $r_0$, Lemma 4 implies that the function $r_0+\alpha(u_2-r_0)$ is a strong supersolution of equation (2.5) for $0<\alpha<1$. Therefore, $u_1=r_0$ almost everywhere by Theorem 4. Case (b) is dealt with similarly. Theorem 5 is proved.
§ 5. Eigenvalue problems5.1. Auxiliary resultWe now dwell on existence conditions of the derivatives of a function $f$ (see (2.3)) mentioned before Theorem 7 in § 2.4. Lemma 5. If the function $f$ (see (2.3)) satisfies the Lipschitz conditions (2.4) and is right concave (convex) at a point $r_0 \in \mathbb{R}$, then at this point it has the right derivative from $L^\infty(X)$ almost everywhere. Similarly, if the function $f$ (see (2.3)) satisfies the Lipschitz conditions (2.4) and is left concave (convex) at a point $r_0 \in \mathbb{R}$, then at this point it has the left derivative from $L^\infty(X)$ almost everywhere. Proof. If the function $f$ is strictly right (left) concave at a point $r_0$, then by the Lipschitz condition (2.8) and the monotonicity property of the difference relations (2.18), the right (left) derivative at the point $r_0$ exists and satisfies the inequality
$$
\begin{equation}
\begin{gathered} \, \frac{f(\,\cdot\,,r)-f(\,\cdot\,,r_0)}{r-r_0}< \frac{\partial f}{\partial r}(\,\cdot\,,r_0+0)\leqslant \mu_0(\max\{|r_0|,|r_1|\}) \\ \biggl(-\mu_0(\max\{|r_0|,|r_1|\})\leqslant \frac{\partial f}{\partial r} (\,\cdot\,,r_0-0)<\frac{f(\,\cdot\,,r)-f(\,\cdot\,,r_0)}{r-r_0}\biggr) \end{gathered}
\end{equation}
\tag{5.1}
$$
almost everywhere for $r_0<r<r_1$ ($r_1<r<r_0$). The case where the function $f$ is strictly left (right) convex at the point $r_0$ is dealt with similarly. Lemma 5 is proved. 5.2. Proof of Theorem 6(a) By the assumption, $\lambda<0$, $w \in L^\infty(X)$ and $\operatorname*{ess\,inf}_{x\in X}w(x)>0$. By definition of the right derivative, there exists $\delta>0$ such that, for all $0<\alpha<\delta$,
$$
\begin{equation*}
\frac{\lambda}{2}\alpha w<f(\,\cdot\,,r_0+\alpha w)- f(\,\cdot\,,r_0)-\alpha w\, \frac{\partial f}{\partial r}(\,\cdot\,,r_0+0)\quad\text{a. e}.
\end{equation*}
\notag
$$
In view of (2.17), (2.16), we have
$$
\begin{equation*}
\begin{aligned} \, 0 &\geqslant \alpha \mathcal{L}(w,v)-\alpha\biggl\langle\frac{\partial f} {\partial r}(\,\cdot\,,r_0+0)w,v\biggl\rangle-\lambda\alpha\langle w,v\rangle \\ &\geqslant \alpha\mathcal{L}(w,v)-\langle f(\,\cdot\,,r_0+\alpha w),v\rangle+ \langle f(\,\cdot\,,r_0),v\rangle+\frac{\lambda}{2}\alpha\langle w,v\rangle- \lambda\alpha\langle w,v\rangle \end{aligned}
\end{equation*}
\notag
$$
for nonnegative functions $v \in C^\infty(X)$ (cf. [3], § 3.2). Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\alpha\mathcal{L}(w,v)+r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\,,r_0+\alpha w),v\rangle- \langle h,dv\rangle_{L^2 (T^* X)} \\ &\qquad\leqslant r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\,,r_0),v\rangle-\langle h,dv\rangle_{L^2(T^* X)}+ \frac{\lambda}{2}\alpha\langle w,v\rangle. \end{aligned}
\end{equation*}
\notag
$$
Hence, using equality (4.1) in Lemma 3 and from the construction of‘$\theta$ (see (2.8)), we have
$$
\begin{equation*}
\mathcal{L}(r_0+\alpha w,v)-\bigl(f(\,\cdot\,,r_0+\alpha w),v\bigr)- \langle h,dv\rangle_{L^2(T^* X)} \leqslant \theta(r_0,v)+ \frac{\lambda}{2}\alpha\langle w,v\rangle
\end{equation*}
\notag
$$
for all $0<\alpha<\beta$. By Remark 3, a number $r_0$ is a subsolution of equation (2.5) if and only if $\theta(r_0,v) \leqslant 0$ for each nonnegative function $v \in C^\infty(X)$. By the assumption, $\operatorname*{ess\,inf}_{x \in X}w(x)>0$ and $\lambda<0$, and hence, for $0<\alpha<\delta$, the function $r_0+\alpha w$ is a subsolution of equation (2.5). Next, by the assumption, the number $N$ is a supersolution of equation (2.5), $N>r_0$, and $w \in L^\infty(X)$, and so, for sufficiently small positive number $\alpha_0$, for $0<\alpha<\alpha_0$ the inequality $r_0+\alpha w \leqslant N$ holds almost everywhere. Now the required result is secured by Theorem 1. Case (b) is dealt with similarly. Theorem 6 is proved. 5.3. Proof of Theorem 8(a) By the assumption, $\operatorname*{ess\,inf}_{x \in X}w(x)>0$, and hence by estimate (5.1), for all $\alpha>0$, we have
$$
\begin{equation}
f(\,\cdot\,,r_0+\alpha w)-f(\,\cdot\,,r_0)< \alpha w\, \frac{\partial f}{\partial r}(\,\cdot\,,r_0+0)\quad\text{a. e}.
\end{equation}
\tag{5.2}
$$
By the assumption (2.17) and (2.16) are satisfied, and hence
$$
\begin{equation*}
\begin{aligned} \, 0 &\leqslant \alpha\mathcal{L}(w,v)- \biggl\langle\frac{\partial f}{\partial r}(\,\cdot\,,r_0+0)w,v\biggr\rangle- \lambda\alpha\langle w,v\rangle \\ &\leqslant \alpha\mathcal{L}(w,v)-\langle f(\,\cdot\,,r_0+\alpha w),v\rangle+ \langle f(\,\cdot\,,r_0),v\rangle-\lambda\alpha\langle w,v\rangle \end{aligned}
\end{equation*}
\notag
$$
for nonnegative functions $v \in C^\infty(X)$, cf. [3], § 3.2. Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\alpha \mathcal{L}(w,v)+r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\,,r_0+\alpha w),v\rangle-\langle h,dv\rangle_{L^2(T^* X)} \\ &\qquad\geqslant r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\,,r_0),v\rangle-\langle h,dv\rangle_{L^2(T^* X)}+ \lambda\alpha\langle w,v\rangle, \end{aligned}
\end{equation*}
\notag
$$
and hence, by equality (4.1) in Lemma 3 and from the construction of $\theta$ (see (2.8)), we get
$$
\begin{equation*}
\mathcal{L}(r_0+\alpha w,v)-\langle f(\,\cdot\,,r_0+\alpha w),v\rangle- \langle h,dv\rangle_{L^2(T^* X)} \geqslant \theta(r_0,v)+\lambda\alpha\langle w,v\rangle;
\end{equation*}
\notag
$$
moreover, since inequality (5.2) is strict, for $v=1$, we have
$$
\begin{equation*}
\mathcal{L}(r_0+\alpha w,1)-\langle f(\,\cdot\,,r_0+\alpha w),1\rangle> \theta(r_0,1)+\lambda\alpha\langle w,1\rangle.
\end{equation*}
\notag
$$
By Remark 3, the number $r_0$ is a supersolution of equation (2.5) if and only if $\theta(r_0,v)\geqslant 0$ for each nonnegative function $v \in C^\infty(X)$, and since, by the assumption, $\operatorname*{ess\,inf}_{x \in X}w(x)>0$ and $\lambda \geqslant 0$, it follows that, for $\alpha > 0$, the function $r_0+\alpha w$ is a strong supersolution of equation (2.5). Next, by assertion (a) of Theorem 2, the solution $u$ is continuous, and so, for sufficiently large positive number $A$, the estimate $r_0 \leqslant u \leqslant r_0+Aw$ holds almost everywhere, from which in view of Theorem 4 we get the required result. Case (b) is dealt with similarly. Theorem 8 is proved. The author is grateful to A. A. Davydov for posing the problem and useful discussions.
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\Bibitem{Tun22} |
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