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This article is cited in 1 scientific paper (total in 1 paper) Semiregular solutions of elliptic boundary-value problems with discontinuous nonlinearities of exponential growth V. N. Pavlenkoa, D. K. Potapovb a Chelyabinsk State University, Chelyabinsk, Russia b Saint Petersburg State University, St. Petersburg, Russia
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     § 1. IntroductionIn a bounded domain $\Omega\subset{\mathbb R}^2$ of class $C^{1,1}$ we consider an elliptic boundary-value problem of the form The differential operator
$$
\begin{equation}
Lu(x)\equiv-\sum_{i,j=1}^2(a_{ij}(x)u_{x_i})_{x_j}+c(x)u(x)
\end{equation}
\tag{1.3}
$$
is uniformly elliptic in $\Omega$ with ellipticity constant $\chi$, $a_{ij}\in C^1(\overline\Omega)$, $a_{ij}(x)=a_{ji}(x)$, $c\in C(\overline\Omega)$ and $c(x)\geqslant 0$ in $\Omega$. The function $g(x,u)$ is superpositionally measurable on $\Omega\times\mathbb R$, that is, for any measurable function $u(x)$ on $\Omega$ the composition $g(x,u(x))$ is measurable. The function $g(x,\cdot\,)$ is assumed to have finite one-sided limits at any point $u\in\mathbb R$ for almost all $x\in\Omega$; we denote them by $g(x,u-)$ and $g(x,u+)$ (the limits from the left and right, respectively). In addition, the nonlinearity $g(x,u)$ admits a growth at infinity of order $|u|^\alpha \exp(A|u|^\tau)$ with respect to the phase variable $u$, where the constants $\alpha$, $A$, and $\tau$ are positive and $\tau<2$. A weak solution of (1.1), (1.2) is a function $u(x)$ in the space $\mathring{W}^1_2(\Omega)$ such that for any $v\in\mathring{W}^1_2(\Omega)$
$$
\begin{equation*}
\sum_{i,j=1}^2\int_\Omega a_{ij}(x)u_{x_i}v_{x_j}\,dx+\int_\Omega c(x)u(x)v(x)\,dx =\int_\Omega g(x,u(x))v(x)\,dx.
\end{equation*}
\notag
$$
A weak solution of problem (1.1), (1.2) is called semiregular if the value $u(x)$ is a point of continuity of the function $g(x,\cdot\,)$ for almost all $x\in\Omega$. We give a statement of the main result of our investigations. 1) for almost all $x\in\Omega$ the inequality $g(x,u-)\leqslant g(x,u+)$ holds and
$$
\begin{equation*}
g(x,u)\in [g(x,u-),g(x,u+)] \quad \forall\, u\in\mathbb R;
\end{equation*}
\notag
$$
2) there are positive constants $b$, $\alpha$, $A$, and $\tau$, $\tau<2$, such that for almost all $x\in\Omega$
$$
\begin{equation}
|g(x,u)|\leqslant a(x)+b|u|^\alpha \exp(A|u|^\tau) \quad \forall\, u\in\mathbb R,
\end{equation}
\tag{1.4}
$$
where $a\in L_N(\Omega)$, and $L_N(\Omega)$ is the Orlicz class of measurable functions on $\Omega$ that is associated with the $N$-function $N(u)$, the complementary function of an $N$-function $M(u)=|u|^{\alpha+2}\exp(|u|^s)$, $s\in (\tau,2)$ (see § 2.1 below); 3) for almost all $x\in\Omega$
$$
\begin{equation*}
\int_0^ug(x,s)\,ds\leqslant \frac{ku^2+d(x)|u|^\theta+d_1(x)}2 \quad \forall\, u\in\mathbb R,
\end{equation*}
\notag
$$
where $k<\chi/\|P_1\|^2$, $\chi$ is the ellipticity constant of the differential operator $L$ defined by (1.3), $P_1$ is the embedding operator of $\mathring{W}^1_2(\Omega)$ in $L_2(\Omega)$, $d\in L_{2/(2-\theta)}(\Omega)$, ${0<\theta<2}$ and $d_1$ is an integrable function on $\Omega$. Then problem (1.1), (1.2) has a weak semiregular solution $u(x)$ in $\mathring W_2^1(\Omega)$. Remark 1. In the proof of Theorem 1, we show that the integral $\displaystyle\int_\Omega \!g(x,u(x))v(x)\,dx$ in the definition of a weak solution of (1.1), (1.2) exists under the conditions of this theorem for any $u,v\in \mathring{W}^1_2(\Omega)$. Remark 2. Condition 3) in Theorem 1 is a hard constraint on a nonlinearity $g(x,u)$ that is of exponential growth at infinity. It is not fulfilled if $g(x,u)\leqslant 0$ on $\Omega\times (-\infty,-u_0]$ or $g(x,u)\geqslant 0$ on $\Omega\times [u_0,+\infty)$ for some $u_0>0$ and this function has exponential growth as $u\to -\infty$ or $u\to +\infty$, respectively. However, if for some ${u_0>0}$ the function $g(x,u)$ is bounded on $\Omega\times [-u_0,u_0]$, positive on ${\Omega\times (-\infty,-u_0]}$ and negative on ${\Omega\times [u_0,+\infty)}$, then condition 3) in Theorem 1 holds for it. For example, the following function is of this type: The existence problem for weak solutions of elliptic boundary-value problems with discontinuous nonlinearities of exponential growth was considered in a number of works. We note the papers [1]–[6], which are typical for this area. In [1] a quasilinear elliptic-type equation with homogeneous Dirichlet boundary condition of the form
$$
\begin{equation}
-\operatorname{div}(a(x,\nabla u))=\lambda h(x)\exp(\alpha_0|u|^{n/(n-1)})+f(x,u), \qquad x\in\Omega,
\end{equation}
\tag{1.5}
$$
was considered in a bounded domain $\Omega\subset{\mathbb R}^n$ ($n\geqslant 2$) with smooth boundary $\partial\Omega$. It was assumed that $a(x,t)$ satisfies the Leray-Lions conditions of growth, coercivity and monotonicity and there exists a Carathéodory function $A(x,t)$ such that $a(x,t)=\nabla_t A(x,t)$. The function $f(x,s)$ is superpositionally measurable on $\Omega\times\mathbb R$ (a function of this type can have discontinuities with respect to $s$) and nondecreasing with respect to $s$ on $\mathbb R$; $f(x,0)=0$ on $\Omega$; $f(x,s)$ can have exponential critical growth in $s$, that is, there is $\alpha_0>0$ such that $\lim_{|s|\to\infty}(|f(x,s)|/\exp(\alpha|s|^{n/(n-1)}))=0$ if $\alpha>\alpha_0$ and this limit is $+\infty$ if $\alpha<\alpha_0$. The parameter $\lambda$ is positive; $h\in L_q(\Omega)$, $q>1$, $h(x)\geqslant 0$ almost everywhere on $\Omega$ and $h(x)>0$ on a set of nonzero measure in $\Omega$. A weak solution of problem (1.5), (1.6) is a function $u\in\mathring{W}^1_n(\Omega)$ such that
$$
\begin{equation*}
\int_\Omega a(x,\nabla u)\cdot\nabla v\,dx\,{=}\,\lambda\int_\Omega h(x)\exp(\alpha_0|u|^{n/(n-1)})v\,dx+\int_\Omega f(x,u)v\,dx \quad\! \forall\, v\,{\in}\,\mathring{W}^1_n(\Omega).
\end{equation*}
\notag
$$
It was established in [1] that for sufficiently small $\lambda$ a nonnegative weak solution exists under the above assumptions. The proof reduces to verifying the conditions of the general Carl-Heikkilä theorem on the existence of a fixed point for a monotone operator in a semiordered Banach space. We present its statement. Theorem 2 (see [7], Corollary 2.2). Let $E$ be a Banach semilattice and let the space $E$ be reflexive. Then any monotone mapping $S\colon B\to B$, where $B\subset E$ is an arbitrary ball, has a fixed point. A Banach space $E$ equipped with a partial order relation $\leqslant$ is called a Banach semilattice if $\sup\{x,y\}$ and $\inf\{x,y\}$ exist for any $x,y\in E$ and $\|x^\pm\|\leqslant \|x\|$ for any $x\in E$. Monotonicity in Theorem 2 is understood in the sense of semiordering: ${Sx\leqslant Sy}$ for any $x$, $y\in B$ such that $x\leqslant y$. Here $x^+=\sup\{0,x\}$ and ${x^-=-\inf\{0,x\}}$. Note that the space $\mathring{W}^1_n(\Omega)$ with the natural partial order relation satisfies the assumptions of Theorem 2. In the case of the operator statement of problem (1.5), (1.6), the Trudinger-Moser inequality (see [8]) and theorems on the embedding of $\mathring{W}^1_n(\Omega)$ in Orlicz spaces (see [9]) are actually used. In [2] the existence of a weak solution of the equation
$$
\begin{equation}
-\operatorname{div} (|\nabla u|^{n-2}\nabla u)+V(x)|u|^{n-2}u=g(x,u)+\lambda h(x)
\end{equation}
\tag{1.7}
$$
in ${\mathbb R}^n$ ($n\geqslant 2$) was established for sufficiently small values of the positive parameter $\lambda$. The potential $V(x)$ can change sign; the nonlinearity $g(x,u)$, which is nondecreasing in $u$, can have critical exponential growth and be discontinuous with respect to $u$. Solutions are considered in the space
$$
\begin{equation*}
X=\biggl\{u\in W^1_n({\mathbb R}^n)\colon \int_{{\mathbb R}^n}V^+(x)|u|^n\,dx<\infty\biggr\}
\end{equation*}
\notag
$$
with the norm
$$
\begin{equation*}
\|u\|=\biggl(\int_{{\mathbb R}^n}\bigl (|\nabla u|^n+V^+(x)|u|^n\bigr)\,dx\biggr)^{1/n},
\end{equation*}
\notag
$$
where $V^+=\max\{V,0\}$. A weak solution of equation (1.7) is a function $u\in X$ such that
$$
\begin{equation*}
\int_{{\mathbb R}^n}|\nabla u|^{n-2}\nabla u\nabla v\,dx+\int_{{\mathbb R}^n}V(x)|u|^{n-2}uv\,dx= \int_{{\mathbb R}^n}g(x,u)v\,dx+\lambda\langle h,v\rangle \quad \forall\, v\in X,
\end{equation*}
\notag
$$
where $\langle h,v\rangle$ is the value of $h\in X^*$ for $v\in X$ ($X^*$ is the dual space of $X$). The existence theorem for weak solutions was proved in [2], just as in [1], by verifying that the conditions of Theorem 2 hold in the space $X$ with the use of a version of the Trudinger-Moser inequality for the whole space. In [3] a generalization of (1.7) in ${\mathbb R}^n$ was considered. It was obtained by replacing the differential part in (1.7) by $-\operatorname{div}(a(x,\nabla u))$ and the nonlinearity $g(x,u)$ by $Q(x)f(u)$, where $Q(x)$ can be unbounded with respect to $x$ and the nondecreasing function $f(u)$ can be discontinuous and have critical exponential growth. Here $a(x,t)$ satisfies the same conditions as in [1], for $\Omega$ replaced by ${\mathbb R}^n$. A sufficient condition for a weak solution to exist in the space $X$ (for the definition of $X$, see above) for sufficiently small values of the parameter $\lambda$ was derived. Like in [1] and [2], the proof reduced to verifying the assumptions of Theorem 2 for the operator statement of problem (1.7) in $X$. An analogue of the equation in [3] with monotone nonlinearity of critical exponential growth was considered in [4] on a compact Riemannian manifold without boundary and in [5] on a noncompact Riemannian manifold of dimension $n\geqslant 2$. By Theorem 2, existence theorems for weak solutions were obtained in a space defined similarly to $X$. In [6] the variational approach was used. That paper considers the inclusion
$$
\begin{equation}
-\Delta u+V(x)u-\varepsilon h(x)\in\partial_t F(x,u), \qquad x\in {\mathbb R}^2,
\end{equation}
\tag{1.8}
$$
where $\varepsilon > 0$; $V$ is a continuous function such that $V(x) \geqslant V_0$ on ${\mathbb R}^2$, where $V_0$ is a positive constant, and $V^{-1} \in L_1({\mathbb R}^2)$; $h \in (W^1_2({\mathbb R}^2))^*$ and $\displaystyle 0 < \int_{{\mathbb R}^2}h(x)\,dx < + \infty$; $\displaystyle F(x,t)=\int_0^t f(x,s)\,ds$, $f(x,s)$ is a discontinuous function of exponential critical growth with respect to $s$; $\partial_t F(x,t)$ is the Clarke generalized derivative of $F(x,t)$ with respect to $t$. It is assumed that $\underline f(x,t):=\lim_{\eta\to t}\inf f(x,\eta)$ and $\overline f(x,t):=\lim_{\eta\to t}\sup f(x,\eta)$ are superpositionally measurable on ${\mathbb R}^2\times\mathbb R$ and there exists $t_0\geqslant 0$ such that $f(x,t)=0$ if $t<t_0$ and $f(x,t)>0$ if $t>t_0$ for any $x\in\mathbb R^2$. Problem (1.8) was considered in the Hilbert space
$$
\begin{equation*}
E=\biggl\{u\in W^1_2({\mathbb R}^2)\colon \int_{{\mathbb R}^2}V(x)u^2(x)\,dx<+\infty\biggr\}
\end{equation*}
\notag
$$
with inner product
$$
\begin{equation*}
(u,v)=\int_{{\mathbb R}^2}\bigl(\nabla u\nabla v+V(x)u(x)v(x)\bigr)\,dx.
\end{equation*}
\notag
$$
A generalized weak solution of problem (1.8) is a function $u\in E$ such that there exists $z(x)$ lying in the intervals $ [\underline f(x,u(x)),\overline f(x,u(x))]$ almost everywhere on ${\mathbb R}^2$ such that
$$
\begin{equation*}
(u,v)-\int_{{\mathbb R}^2}z(x)v(x)\,dx-\varepsilon\int_{{\mathbb R}^2}h(x)v(x)\,dx=0
\end{equation*}
\notag
$$
for any $v\in E$ and $\operatorname{mes}\{x\in {\mathbb R}^2\colon u(x)>t_0\}\neq 0$. The following restrictions were additionally imposed in [6]:
Note that the above conditions on the nonlinearity are satisfied if for all $t\in\mathbb R$. Here $a\geqslant 0$, $H(s)=0$ for $s\leqslant 0$ and $H(s)=1$ for $s>0$ (the Heaviside step function).Under the above conditions the existence of two solutions of inclusion (1.8) was proved in [6] for sufficiently small $\varepsilon$ and $t_0$ and sufficiently large $\mu$ in condition 3). Ekeland’s variational principle and the mountain pass theorem for nondifferentiable functionals were used in the proof. Semiregular solutions of elliptic-type equations with discontinuous nonlinearities were introduced by Krasnosel’skii and Pokrovskii [10]. In [11] they deduced an existence theorem for a semiregular solution of an elliptic boundary-value problem with discontinuous nonlinearity using the method of upper and lower solutions. The present authors obtained more general results in [12]. Abstract theorems in this area were proved in [13]. We mention the most recent papers [14]–[20] concerning semiregular solutions. Note that equations with nonlinearities of exponential growth were not considered in [10]–[20]. This paper presents a sufficient existence condition for a semiregular solution of problem (1.1), (1.2) with a discontinuous nonlinearity admitting an exponential growth with respect to the phase variable (Theorem 1) to exist. We do not assume that the nonlinearity $g(x,s)$ is nondecreasing with $s$ (a key requirement in [1]–[5]) and that $g(x,s)=0$ for $s<t_0$ and $g(x,s)>0$ for $s>t_0$ for some $t_0$ (one of the assumptions in [6]). In addition, in the proof of Theorem 1 we use the variational approach based on the concept of a quasipotential operator (see, for example, [21]–[23]), unlike the traditional approach, which uses the Clarke generalized derivative. One of the most recent papers where it was used was [20]. § 2. Preliminary information2.1. Orlicz spacesThe necessary notions and facts concerning Orlicz spaces have been borrowed from [24]. A continuous convex function $M(u)$ on $\mathbb R$ is called an $N$-function if it is even and satisfies the conditions $\lim_{u\to 0}M(u)/u\,{=}\,0$ and $\lim_{u\to\infty}M(u)/u\,{=}\,{+}\infty$. Examples of $N$-functions are: $\varphi(u)=\exp(u^2)-1$ and $\psi(u)=|u|^{\alpha+2}\exp(|u|^s)$, where $\alpha$ and $s$ are positive constants. The complementary function of an $N$-function $M(u)$ is its Legendre transform $N(u)=\max\{|u|t-M(t)\colon t\geqslant 0\}$. Note that $N(u)$ is also an $N$-function. The functions $M$ and $N$ are called mutually complementary or conjugate. An $N$-function $M(u)$ is said to satisfy the $\Delta_2$-condition if $M(lu)\leqslant m(l)M(u)$ for sufficiently large $u$, where $l$ can be any number greater than 1. Note that functions satisfying the $\Delta_2$-condition grow not faster than power functions. In particular, the functions $\varphi(u)$ and $\psi(u)$ defined above do not satisfy the $\Delta_2$-condition. The natural question arises as to how we can verify whether the complementary function $N(u)$ of $M(u)$ satisfies the $\Delta_2$-condition. It turns out that the existence of $l>1$ and $v_0\geqslant 0$ such that $M(v)\leqslant M(lv)/(2l)$ for all $v\geqslant v_0$ is necessary and sufficient for this (see [24], Ch. I, § 4, Theorem 4.2). It immediately follows that the complementary functions of $\varphi(u)$ and $\psi(u)$ satisfy the $\Delta_2$-condition. To prove Theorem 1 we need the inequality where $M$ and $N$ are mutually complementary functions, while $M^{-1}$ and $N^{-1}$ are the inverse functions.Let $\Omega$ be a bounded domain in ${\mathbb R}^n$ and $M(u)$ be an $N$-function. The Orlicz class associated with $M(u)$ is the set
$$
\begin{equation*}
\begin{aligned} \, L_M(\Omega)&=\biggl\{\text{measurable functions } u\colon \Omega\to {\mathbb R} \\ &\qquad\qquad\text{such that } \rho(u,M)=\int_\Omega M(u(x))\,dx<\infty\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
Functions that differ on a zero measure set are considered to be indistinguishable in this case. The notation $L_M$ is used instead of $L_M(\Omega)$ when this cannot lead to misunderstanding. An Orlicz class is a convex set containing all bounded measurable functions. Its elements are integrable functions; however, not any integrable function belongs to $L_M$. The class $L_M$ is a linear space if and only if $M(u)$ satisfies the $\Delta_2$-condition. Let $M(u)$ and $N(v)$ be mutually complementary $N$-functions. We set
$$
\begin{equation*}
\begin{aligned} \, L_M^*(\Omega)&=\biggl\{\text{measurable functions } u\colon\Omega\to{\mathbb R} \\ &\qquad\qquad\text{such that } (u,v)=\int_\Omega u(x)v(x)\,dx<\infty\ \forall\, v\in L_N(\Omega)\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
In this case, like in the definition of Orlicz classes, functions that differ on a zero measure set are considered to be indistinguishable. The notation $L_M^*$ is used instead of $L_M^*(\Omega)$ when this cannot lead to misunderstanding. The set $L_M^*(\Omega)$ is a linear space and coincides with the span of $L_M(\Omega)$. It is endowed with the Orlicz norm
$$
\begin{equation*}
\|u\|_M=\sup\biggl\{\biggl|\int_\Omega u(x)v(x)\,dx\biggr|\colon \rho(v,N)\leqslant 1\biggr\}.
\end{equation*}
\notag
$$
This normed space is complete; it is denoted by $L_M^*(\Omega)$ and called the Orlicz space associated with the $N$-function $M(u)$. The following two inequalities are important for further considerations:
$$
\begin{equation}
\|u\|_M\leqslant\rho(u,M)+1\quad \forall\, u\in L_M;
\end{equation}
\tag{2.2}
$$
if $\|u\|_M\leqslant 1$, then $u\in L_M$ and The Luxemburg norm, which is equivalent to the Orlicz norm, can be introduced in the space $L_M^*$ by the formula
$$
\begin{equation*}
\|u\|_{(M)}=\inf\biggl\{m\colon \rho\biggl(\frac um,M\biggr)\leqslant 1\biggr\}.
\end{equation*}
\notag
$$
It is related to the Orlicz norm by The closure of the set of bounded functions in $L_M^*$ is denoted by $E_M$. If $M(u)$ satisfies the $\Delta_2$-condition, then $E_M=L_M^*=L_M$. In the general case the dual space $E_M^*$ of $E_M$ coincides with $L_N^*$ endowed with the Luxemburg norm, where $N(u)$ is the complementary function of $M(u)$. We switch to Trudinger’s embedding theorems for Orlicz spaces. An $N$-function $M_1(u)$ is said to grow significantly faster than an $N$-function $M(u)$ if $\lim_{u\to\infty}M(\lambda u)/M_1(u)=0$ for any positive $\lambda$. In this case, following [9], we write $M\prec M_1$. It is straightforward to verify that for the functions $\varphi(u)$ and $\psi(u)$ defined above we have $\psi\prec\varphi$. The following theorem is true. Theorem 3 (see [9], Theorem 2). Let $\Omega$ be a bounded domain in ${\mathbb R}^n$ satisfying the cone condition. Then the Sobolev space $W^k_p(\Omega)$, where $n=kp$, is continuously embedded in the Orlicz space $L_\Phi^*(\Omega)$, where $\Phi(t)=\exp(|t|^{n/(n-1)})-1$. In addition, for any $N$-function $\gamma(t)$ such that $\gamma\prec\Phi$ this embedding in $L_\gamma^*(\Omega)$ is compact. We need this theorem in the two-dimensional case, for $k=1$ and $p=2$. 2.2. Quasipotential operatorsLet $E$ be a real Banach space and let $E^*$ be the dual space of $E$. We let $\langle z,x \rangle$ denote the value of the functional $z\in E^*$ at an element $x\in E$. In what follows, an operator $T$ acts from $E$ to $E^*$. A functional $f\colon E\to\mathbb R$ is said to be Gateaux differentiable at a point $x\in E$ if there is $y\in E^*$ such that the limit $\lim_{t\to 0}(f(x+th)-f(x))/{t}=\langle y,h \rangle$ exists for each $h\in E$. In this case $y$ is called the Gateaux derivative of $f$ at $x$ and is denoted by $f'(x)$. An operator $T$ is said to be potential if there exists a Gateaux differentiable functional $f$ on $E$ such that $f'(x)=Tx$ for any $x\in E$. A functional of this type is called a potential of the operator $T$. An operator $T$ is said to be radially integrable if for any $x$, $h\in E$ the function $\langle T(x+th),h \rangle$ is integrable on $[0,1]$. An operator $T$ is said to be radially continuous at a point $x\in E$ if for any $h\in E$ we have $\lim_{t\to 0}\langle T(x+th),h \rangle=\langle Tx, h\rangle$. An element $x\in E$ is called a point of discontinuity of an operator $T$ if there is $h\in E$ such that either $\lim_{t\to 0}\langle T(x+th), h\rangle$ does not exist or ${\lim_{t\to 0}\langle T(x+th), h\rangle\neq\langle Tx, h\rangle}$. A radially integrable operator $T$ is said to be quasipotential (see [25], Ch. 5, § 17, Definition 17.15) if there exists a functional $f\colon E\,{\to}\,{\mathbb R}$ such that
$$
\begin{equation}
f(x+h)-f(x)=\int^1_0\langle T(x+th), h\rangle\, dt \quad \forall\, x, h\in E.
\end{equation}
\tag{2.4}
$$
In this case $f$ is called a quasipotential of $T$. Note that if an operator $T$ is potential and radially continuous on $E$ and $f$ is a potential of it, then for all $x , h\in E$ we have
$$
\begin{equation*}
f(x+h)-f(x)=\int_0^1\frac{d}{dt}f(x+th)\,dt=\int_0^1\langle T(x+th),h\rangle \,dt.
\end{equation*}
\notag
$$
However, a quasipotential operator can be discontinuous. For example, if $E=\mathbb R$, $Tx=\operatorname{sgn}x$, and $f(x)=|x|$, then (2.4) holds, but $T$ is discontinuous at $x=0$. To state the variational principle to be used in the proof of Theorem 1 we need a further important notion. An element $x\in E$ is called a regular point of an operator $T$ if for some $h\in E$ we have $\lim_{t\to +0}\sup\langle T(x+th), h\rangle<0$. The following theorem is valid. Theorem 4 (see [23], Theorem 1). Let $x\in E$ be a minimum point of a quasipotential $f$ of a locally bounded operator $T\colon E\to E^*$ whose points of discontinuity are regular. Then $x$ is a radial continuity point of $T$ and $Tx=0$. We present a sufficient condition from [22] for the regularity of points of discontinuity of an operator $T$: if for any $x$, $h\in E$ the nonpositive limit ${\lim_{t\to +0}\langle T(x+th)-Tx,h\rangle}$ exists, then all points of discontinuity of $T$ are regular. Now we prove this. Let $x$ be a point of discontinuity of $T$. By assumption $\lim_{t\to +0}\langle T(x+th),h\rangle$ exists for any $h\in E$ and does not exceed $\langle Tx,h\rangle$. Since $x$ is a point of discontinuity, there exists $z\in E$ such that $\lim_{t\to +0}\langle T(x+tz),z\rangle<\langle Tx,z\rangle$. Otherwise $x$ is a point of radial continuity of $T$. Furthermore, if $\langle Tx,z\rangle\leqslant 0$, then $\lim_{t\to +0}\langle T(x+tz),z\rangle<0$ because of the last inequality; hence $z$ is a regular point of the operator $T$. If $\langle Tx,z\rangle>0$, then $\lim_{t\to +0}\langle T(x+t(-z)),-z\rangle$ exists and satisfies the inequalities $\lim_{t\to +0}\langle T(x+t(-z)),-z\rangle\leqslant \langle Tx,-z\rangle =-\langle Tx,z\rangle<0$. Therefore, $x$ is a regular point of $T$. We give another result, which is useful for further consideration. Theorem 5 (see [26]). Let $T=T_1+T_2$, where $T_1$ is a monotone operator (that is, $\langle T_1u-T_1v,u-v\rangle\geqslant 0$ for any $u,v\in E$), $T_2$ is a compact operator (that is, it maps bounded sets in $E$ to precompact sets in $E^*$), and both $T_i$, $i=1,2$, are quasipotential. Then a quasipotential $f$ of $T$ is weakly semicontinuous from below on $E$ (that is, for any $x\in E$ and any sequence $(x_n)\subset E$ weakly converging to $x$ the inequality $\lim_{n\to\infty}\inf f(x_n)\geqslant f(x))$ holds. § 3. Proof of Theorem 1Let $E=\mathring{W}^1_2(\Omega)$ and $E^*=W^{-1}_2(\Omega)$. The space $E$ is a reflexive Hilbert space with inner product
$$
\begin{equation*}
(u,v)=\int_\Omega \nabla u\cdot\nabla v\,dx \quad \forall\, u,v\in E,
\end{equation*}
\notag
$$
which induces the norm $\|u\|=\sqrt{(u,u)}$. We define operators $T_i$, $i=1,2$, on $E$ by
$$
\begin{equation*}
\langle T_1u,v\rangle=\sum_{i,j=1}^2\int_\Omega a_{ij}(x)u_{x_i}v_{x_j}\,dx+\int_\Omega c(x)u(x)v(x)\,dx,
\end{equation*}
\notag
$$
for any $u$, $v\in E$. In the proof of Theorem 1 we adhere to the following five-part plan. 1) We establish that the operator $T_1$ is bounded, quasipotential, and monotone. 2) We prove that the operator $T_2$ is quasipotential and compact. By Theorem 5 it follows from parts 1) and 2) that the quasipotential $J(u)$ of the operator $T=T_1-T_2$ is weakly semicontinuous from below on $E$. 3) We prove that the functional $J(u)$ is coercive, that is, $\lim_{\|u\|\to\infty}\!J(u)\,{=}\,{+}\infty$. Since a functional in a reflexive Banach space that is weakly semicontinuous from below and coercive attains its global minimum (see [25], Ch. III, § 9, Remark 9.1), there exists $u_0\in E$ such that $J(u_0)=\inf\{J(u)\colon u\in E\}$. 4) We prove that points of discontinuity of $T$ are regular. To do this we verify that the conditions in the regularity test for points of discontinuity of the operator $T$ are fulfilled. The statement and proof of this test are presented in § 2.2. Theorem 4 implies that $u_0$ is a point of radial continuity of the operator $T$ and $Tu_0=0$. By the definition of $T$, a function $z(x)\in E$ is a weak solution of problem (1.1), (1.2) if and only if $Tz=0$. Therefore, $u_0(x)$ is a weak solution of (1.1), (1.2). 5) We prove that, as the operator $T$ is radially continuous at the point $u_0$, the set $U=\{x\in\Omega\colon u_0(x)$ is a point of discontinuity of the function $g(x,\cdot\,)\}$ has measure zero. Thus, $u_0(x)$ is a weak semiregular solution of problem (1.1), (1.2). This completes the proof of Theorem 1. We switch to implementing the above plan of the proof. The operator $T_1$ is linear, bounded and self-adjoint (that is, $\langle T_1 u,v\rangle=\langle T_1v,u\rangle$). Since $c(x)\geqslant 0$ on $\Omega$, we have
$$
\begin{equation}
\langle T_1u,u\rangle\geqslant\chi \|u\|^2 \quad \forall\, u\in E,
\end{equation}
\tag{3.2}
$$
where $\chi$ is the ellipticity constant of the differential operator $L$. It follows from (3.2), since $T_1$ is linear, that $T_1$ is a monotone operator. This operator is potential, and has a potential $f_1(u)=(1/2)\langle T_1u,u\rangle$ ([25], Ch. II, § 5, Example 5.6). Now we realize part 2) of the plan. We need to establish that the operator $T_2$ acts from $E$ to $E^*$, is compact and quasipotential. First we prove that the Nemytskii operator $G(u)=g(x,u(x))$ maps bounded sets in $E_M(\Omega)$ (where $M(u)$ is the $N$-function in condition 2) of Theorem 1) to bounded sets in $L_N^*(\Omega)$, $N(u)$ is the complementary function of $M(u)$. As noted above, $N(u)$ satisfies the $\Delta_2$-condition. Therefore, $L_N^*=L_N=E_N$. Since $E_M^*=L_N^*$, we have $E_M^*=E_N$ in this case. We fix $r>0$ and show that there exist $a_r(x)\in L_N$ and $b_r>0$ such that for almost all $x\in\Omega$ the right-hand side of (1.4) in condition 2) of Theorem 1 does not exceed $a_r(x)+b_rN^{-1}(M(u/r))$ for any $u\in\mathbb R$. In fact,
$$
\begin{equation*}
\begin{aligned} \, &\lim_{u\to\infty}\frac{M(u/r)}{(|u|/r)|u|^\alpha \exp(A|u|^\tau)}= \lim_{u\to\infty}\frac{|u|^{\alpha+2}\exp(|u/r|^s)}{r^{\alpha+2}|u|^\alpha \exp(A|u|^\tau)}\,\frac{r}{|u|} \\ &\qquad =\lim_{u\to\infty}\frac{|u|\exp(|u/r|^s)}{r^{\alpha+1}\exp(A|u|^\tau)}=+\infty, \end{aligned}
\end{equation*}
\notag
$$
since $s>\tau$ by virtue of condition 2) in Theorem 1. It follows that there exist $a_r(x)\in L_N(\Omega)$ and $b_r>1$ such that
$$
\begin{equation}
a(x)+b|u|^\alpha \exp(A|u|^\tau)\leqslant a_r(x)+\frac{b_rM(u/r)}{|u|/r} \quad \forall\, u\in\mathbb R
\end{equation}
\tag{3.3}
$$
for almost all $x\in\Omega$. Due to (2.1), we have $N^{-1}(M(u/r))\geqslant M(u/r)/(|u|/r)$ for each $ u\in\mathbb R$. This fact and (3.3) yield
$$
\begin{equation*}
a(x)+b|u|^\alpha \exp(A|u|^\tau)\leqslant a_r(x)+b_rN^{-1}\biggl(M\biggl(\frac ur\biggr)\biggr) \quad \forall\, u\in\mathbb R
\end{equation*}
\notag
$$
for almost all $x\in\Omega$. Thus, we have shown that for any $r>0$ there exist $a_r(x)\in L_N(\Omega)$ and $b_r>1$ such that
$$
\begin{equation}
|g(x,u)|\leqslant a_r(x)+b_rN^{-1}\biggl(M\biggl(\frac ur\biggr)\biggr) \quad \forall\, u\in\mathbb R
\end{equation}
\tag{3.4}
$$
for almost all $x\in\Omega$. It follows from this inequality that the Nemytskii operator $G$ is bounded on the ball $B_r=\{u\in E_M\colon \|u\|_M\leqslant r\}$, that is, there is a constant $C_r>0$ such that $\|Gu\|_N\leqslant C_r$ for any $u\in B_r$. In fact, since $N(u)$ is increasing and convex on $\mathbb R$, for almost all $x\in\Omega$ we have
$$
\begin{equation*}
N(g(x,u))\leqslant\frac12N(a_r(x))+\frac12N\biggl(b_rN^{-1}\biggl(M\biggl(\frac ur\biggr)\biggr)\biggr) \quad \forall\, u\in\mathbb R.
\end{equation*}
\notag
$$
Since $N(u)$ satisfies the $\Delta_2$-condition, there exist a number $l(b_r)>0$ and $u_0> 0$ such that $N(b_rN^{-1}(M(u/r)))\leqslant l(b_r)N(N^{-1}(M(u/r)))=l(b_r)M(u/r)$ for ${|u|\,{\geqslant}\, u_0}$. From this we deduce that there exists an integrable function $d_r(x)$ on $\Omega$ such that
$$
\begin{equation}
N(g(x,u))\leqslant d_r(x)+\frac12l(b_r)M\biggl(\frac ur\biggr) \quad \forall\, u\in\mathbb R
\end{equation}
\tag{3.5}
$$
for almost all $x\in\Omega$. It follows that if $u(x)\in E_M$ and $\|u\|_M\leqslant r$, then
$$
\begin{equation*}
\begin{aligned} \, \int_\Omega N(g(x,u(x))\,dx &\leqslant\int_\Omega d_r(x)\,dx+\frac12l(b_r)\int_\Omega M\biggl(\frac{u(x)}r\biggr)\,dx \\ &\leqslant\int_\Omega d_r(x)\,dx+\frac12l(b_r)\biggl\|\frac ur\biggr\|_M \leqslant\int_\Omega d_r(x)\,dx+\frac12l(b_r)=A_r \end{aligned}
\end{equation*}
\notag
$$
by (3.5) (we have used (2.3) since $\|u/r\|_M\leqslant 1$). By virtue of (2.2), we have
$$
\begin{equation}
\|Gu\|_N\leqslant\int_\Omega N(g(x,u(x))\,dx+1\leqslant A_r+1
\end{equation}
\tag{3.6}
$$
if $u\in B_r$. The boundedness of the operator $G$ on the ball $B_r$ is proved. We show that equality (3.1) defines the operator $T_2\colon E\to E^*$ consistently and establish that it is compact. As noted above, $M\prec\varphi$, where $M(u)$ is the $N$-function in condition 2) of Theorem 1 and $\varphi(u)=\exp(u^2)-1$. Then, according to Theorem 3 the embedding of $W^1_2(\Omega)$ in $L_M^*(\Omega)$ is compact. We show that $W_2^1(\Omega)\subset E_M$. Since the bounded domain $\Omega\subset {\mathbb R}^2$ is of class $C^{1,1}$, the set $C^\infty(\overline\Omega)$ is dense in $W_2^1(\Omega)$. The space $E_M$ is the closure of the set of measurable bounded functions on $\Omega$ in the space $L_M^*$. Consequently, $E_M\supset C^\infty(\overline\Omega)$. Since $E_M$ is a closed subset in $L_M^*$, this inclusion implies that $E_M\supset \overline{C^\infty(\overline\Omega)}$, where the right-hand side is the closure of $C^\infty(\overline\Omega)$ in $L_M^*$. Since the embedding of $W_2^1(\Omega)$ in $L_M^*$ is compact, we have ${\overline{C^\infty(\overline\Omega)}\supset W_2^1(\Omega)}$. It follows that $E_M\supset W_2^1(\Omega)$. Since $E$ is a closed subspace in $W_2^1(\Omega)$, the embedding of $E$ in $E_M$ is compact. We denote the embedding operator of $E$ in $E_M$ by $P$. The dual operator $P^*$ embeds $E_M^*=E_N$ in ${E^*=W_2^{-1}(\Omega)}$. It is compact since $P$ is ([27], Ch. 4, § 6, Theorem 3). Owing to the theorem on the general form of a linear bounded functional on $E_M$ ([24], Ch. II, § 14, Theorem 14.2), for arbitrary $u$, $v\in E$ we have
$$
\begin{equation*}
\int_\Omega g(x,u(x))v(x)\,dx=\langle G(Pu),Pv\rangle=\langle P^*GP(u),v\rangle.
\end{equation*}
\notag
$$
We conclude that $T_2u=P^*GP(u)$ for all $u\in E$. This representation of $T_2$ implies its compactness, since $G$ maps bounded sets in $E_M$ to bounded sets in $E_M^*=E_N$ and the embedding operators $P$ and $P^*$ are compact. It remains to prove that the operator $T_2$ is quasipotential. For this purpose it suffices to prove that $G$ is quasipotential. In fact, if $G$ is a quasipotential operator and $f$ is a quasipotential of it, then we have
$$
\begin{equation*}
f(u+h)-f(u)=\int_0^1\langle G(u+th),h\rangle \,dt \quad \forall\, u,h\in E_M.
\end{equation*}
\notag
$$
Then it is true for arbitrary $u$, $h\in E$ that
$$
\begin{equation*}
\begin{aligned} \, &f(Pu+Ph)-f(Pu)=\int_0^1\langle G(Pu+tPh),Ph\rangle\, dt \\ &\qquad =\int_0^1\langle P^*GP(u+th),h \rangle \,dt=\int_0^1\langle T_2(u+th),h\rangle \,dt, \end{aligned}
\end{equation*}
\notag
$$
which means that the operator $T_2$ is quasipotential and $f_2(u)=f(Pu)$ is a quasipotential of it. Now we prove that $G$ is a quasipotential operator. In calculations below we use the following three classical results:
We define a functional $f$ on $E_M$ by the equality
$$
\begin{equation}
f(u)=\int_\Omega dx\int_0^{u(x)}g(x,s)\,ds\quad \forall\, u\in E_M.
\end{equation}
\tag{3.7}
$$
For arbitrary $u$, $h\in E_M$ we have
$$
\begin{equation}
\begin{aligned} \, \notag f(u+h)-f(u) &=\int_\Omega dx\int_{u(x)}^{u(x)+th(x)}g(x,s)\,ds \\ \notag &=\int_\Omega dx\int_0^1\frac{d}{dt}\int_0^{u(x)+th(x)}g(x,s)\,ds\,dt \\ &=\int_0^1dt\int_\Omega g(x,u(x)+th(x))h(x)\,dx=\int_0^1\langle G(u+th),h\rangle \,dt. \end{aligned}
\end{equation}
\tag{3.8}
$$
The integrability of $\psi(x,t)=g(x,u(x)+th(x))h(x)$ on $\Omega\times [0,1]$ follows from Hölder’s inequality (see [24], Ch. II, § 9, Theorem 9.3)
$$
\begin{equation*}
\biggl|\int_\Omega z(x)y(x)\,dx\biggr|\leqslant \|z\|_M\, \|y\|_N
\end{equation*}
\notag
$$
for any $z\in L_M$ and $y\in L_N$ and from estimate (3.6). Note that we can verify that $f$ is finite on $E_M$ by making this calculation in the reverse order for $u=0$. It follows from (3.8) that the operator $G$ is quasipotential; a quasipotential of it is specified by (3.7). As mentioned above, this implies that $T_2$ is a quasipotential operator, and $f_2(u)=f(Pu)$ is a quasipotential of it. The second part of the plan is realized. So, $T=T_1-T_2$ is a quasipotential operator and $J(u)=(1/2)\langle T_1u,u\rangle-f(Pu)$ is a quasipotential of it. By the definition of the operators $T_1$ and $T_2$, a function $z(x)\in E$ is a weak solution of problem (1.1), (1.2) if and only if $Tz=0$. By Theorem 5, since the operator $T_1$ is monotone and $T_2$ is compact, the functional $J(u)$ is weakly semicontinuous from below on $E$. We switch to part 3) of the plan. We prove that the functional $J(u)$ is coercive, that is, $\lim_{\|u\|\to\infty}J(u)=+\infty$. In fact, by condition 3) in Theorem 1, for an arbitrary $u\in E$ we have
$$
\begin{equation}
\begin{aligned} \, \notag J(u) &=\frac12\langle T_1u,u\rangle-f(Pu)\geqslant\frac{\chi}{2}\|u\|^2-\int_\Omega dx\int_0^{u(x)}g(x,s)\,ds \\ \notag &\geqslant\frac{\chi}{2}\|u\|^2-\frac12\int_\Omega \biggl(ku^2(x)+d(x)|u|^\theta+d_1(x)\biggr)\,dx \\ &\geqslant \frac{\chi-k\|P_1\|^2}{2}\|u\|^2-\frac{\|d\|_{2/(2-\theta)}}{2}\|u\|_2^\theta-\frac{\|d_1\|_1}{2}, \end{aligned}
\end{equation}
\tag{3.9}
$$
where $\chi$ is the ellipticity constant of the operator $L$ defined by (1.3), $P_1$ is the embedding operator of $E$ in $L_2(\Omega)$ and $\|\cdot\|_s$ is the norm in $L_s(\Omega)$. By condition 3) in Theorem 1, the constant $\chi-k\|P_1\|^2$ is positive; in addition, $\|u\|_2\leqslant \|P_1\|\,\|u\|$ and $0<\theta<2$. By virtue of (3.9) we derive from this that $J(u)$ is coercive. A weakly lower semicontinuous coercive functional in a reflexive Banach space is bounded below and attains its global minimum (see [25], Ch. III, § 9, Remark 9.1). Therefore, there is $u_0\in E$ such that $J(u_0)=\inf\{J(u)\colon u\in E\}$. We switch to implementing part 4) of the plan. We prove that all points of discontinuity of the operator $T$ are regular. Then, according to Theorem 4, $Tu_0=0$ and $u_0$ is a point of radial continuity of $T$. A sufficient condition for the regularity of points of discontinuity of $T$ is the inequality $\lim_{t\to +0}\langle T(u+th)-Tu,h\rangle\leqslant 0$ $\forall\, u, h\in E$. For any $u$, $h\in E$ and $t\in (0,1)$ we have
$$
\begin{equation*}
\langle T_2(u+th)-T_2u,h\rangle=\int_\Omega g(x,u(x)+th(x))h(x)\,dx-\int_\Omega g(x,u(x))h(x)\,dx.
\end{equation*}
\notag
$$
By condition 1) in Theorem 1, $\lim_{t\to +0}g(x,u(x)+th(x))h(x)$ exists for almost all $x \in \Omega$ and is no smaller than $g(x,u(x))h(x)$. This implies that the following limit exists:
$$
\begin{equation}
\lim_{t\to +0}\langle T_2(u+th)-T_2u,h\rangle\geqslant 0,
\end{equation}
\tag{3.10}
$$
provided that we justify taking the limit under the integral sign. To use Lebesgue’s theorem we need to prove that, given $u(x)$ and $h(x)$ in $E$, there exists an integrable function $\psi(x)$ on $\Omega$ such that almost everywhere on $\Omega$ for any $t\in (0,1)$. We take $r=\bigl\||u|+|h|\bigr\|_M$. As shown above, there exist $a_r(x)\in L_N$ and $b_r>1$ such that (3.4) holds for almost all $x\in\Omega$ and any $u\in\mathbb R$. We set $\psi_1(x)=a_r(x)+b_rN^{-1}(M((|u(x)|+|h(x)|)/r))$, $x\in\Omega$. By virtue of (3.4), taking account of the fact that $N^{-1}$ and $M$ are increasing functions, we see that (3.11) is valid for $\psi(x)=\psi_1(x)|h(x)|$, for any $t\in (0,1)$. Note that $\psi_1(x)\in L_N(\Omega)$ because
$$
\begin{equation*}
N(\psi_1(x))\leqslant d_r(x)+\frac12l(b_r)M\biggl(\frac{|u(x)|+|h(x)|}{r}\biggr)
\end{equation*}
\notag
$$
and $u(x)$, $h(x)\in E_M$. Here $d_r(x)$ and $l(b_r)$ are the same as in (3.5). Since $h\in L_M$, it follows that $\psi(x)$ is integrable on $\Omega$ by Hölder’s inequality. Inequality (3.10) is proved. From (3.10) and the radial continuity of $T_1$ we derive that $\lim_{t\to +0}\langle T(u+th)-Tu,h\rangle\leqslant 0$ for any $u$, $h\in E$. The regularity of points of discontinuity of the operator $T$ is established. According to Theorem 4, $Tu_0=0$ and $u_0$ is a point of radial continuity of $T$. To complete the proof of Theorem 1 it suffices to prove that if $T$ is radially continuous at the point $u_0$, then the set $U=\{x\in \Omega\colon u_0(x) \text{ is a point of discontinuity of} \text{the function } g(x,\cdot\,)\}$ has measure zero (part 5) of the plan). In view of condition 1) in Theorem 1 the set $U$ coincides with $\{x\in\Omega\colon g(x,u_0(x)-)<g(x,u_0(x)+)\}$ up to a nullset. Assume that $\operatorname{mes} U\neq 0$. Then there exist $\varepsilon>0$ and $\delta>0$ such that $U(\varepsilon)=\{x\in \Omega\colon g(x,u_0(x)+)-g(x,u_0(x)-)>\varepsilon\}$ has measure $\delta$. We set $r=\|u_0\|_M+\|l\|_M$ ($l\equiv 1$ on $\Omega$) and $\widehat{\psi}(x)=a_r(x)+b_rN^{-1}(M((|u_0(x)|+1)/r))$, $x\in\Omega$, where $a_r(x)$ and $b_r$ are from (3.4). Like in the proof of the regularity of points of discontinuity of $T$, we can prove that $\widehat{\psi}(x)\in L_N$ and thus $\widehat{\psi}(x)$ is integrable on $\Omega$. Since the Lebesgue integral is absolutely continuous (see [27], Ch. V, § 5, Theorem 5), there exists $\nu>0$ such that if $\omega$ is a measurable subset of $\Omega$ and $\operatorname{mes}\omega\leqslant\nu$, then
$$
\begin{equation*}
\int_\omega\widehat{\psi}(x)\,dx<\frac{\varepsilon\delta}{8}.
\end{equation*}
\notag
$$
The set $U(\varepsilon)\subset\Omega$ is measurable. Therefore, there exist a closed set $F\subset U(\varepsilon)$ and an open set $H \supset F$ with closure $\overline H\subset\Omega$ such that $\operatorname{mes}F>\operatorname{mes} U(\varepsilon)/2=\delta/2$ and $\operatorname{mes}(H\setminus F)<\nu$ (see [28]). Let $h \in C^\infty(\overline{\Omega})$ be equal to 1 on $F$, to 0 outside $H$, and ${0\leqslant h(x)\leqslant 1}$ for $x\in H\setminus F$ (such a function exists by virtue of the lemma in [29], Ch. 14, § 2). Note that $h\in E$. In view of (3.4) and the fact that the functions $N^{-1}$ and $M$ are increasing we see that for almost all $x\in\Omega$, for $t\in (-1,1)$. Owing to Lebesgue’s dominated convergence theorem, from condition 1) in Theorem 1 and (3.12) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\lim_{t\to +0}\int_\Omega g\bigl(x,u_0(x)+th(x)\bigr)h(x)\,dx \\ &\qquad=\int_F g(x,u_0(x)+)\,dx+\int_{H\setminus F}\lim_{t\to +0}g\bigl(x,u_0(x)+th(x)\bigr)h(x)\,dx, \\ &\lim_{t\to -0}\int_\Omega g\bigl(x,u_0(x)+th(x)\bigr)h(x)\,dx \\ &\qquad=\int_F g(x,u_0(x)-)\,dx+\int_{H\setminus F}\lim_{t\to -0}g\bigl(x,u_0(x)+th(x)\bigr)h(x)\,dx. \end{aligned}
\end{equation*}
\notag
$$
Consequently, the difference between the left-hand sides of the last two equalities is greater than
$$
\begin{equation*}
\frac{\varepsilon\delta}{2}-2\frac{\varepsilon\delta}{8}=\frac{\varepsilon\delta}{4}>0,
\end{equation*}
\notag
$$
which contradicts the radial continuity of $T$ at $u_0$, since $T_1$ is a radially continuous operator. Theorem 1 is proved.
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\Bibitem{PavPot22} |
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