| Izvestiya: Mathematics |
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One class of quasilinear elliptic type equations with discontinuous nonlinearities V. N. Pavlenkoa, D. K. Potapovb a Chelyabinsk State University b Saint Petersburg State University
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     § 1. Introduction, statement of the problem, and the main resultsIn a given bounded C1,1-smooth domain Ω⊂Rn (see [1]), consider the following boundary-value problem with homogeneous boundary Dirichlet condition for a quasilinear elliptic type equation with nonnegative parameter λ and discontinuous nonlinearity g(x,u) of the following form: It is assumed that the coefficients of the differential operator L satisfy the following conditions: (i1) aij(x,t)=aji(x,t) on ¯Ω×R, and there exists a nonincreasing positive on R+ function χ(t) such that, for any (x,t)∈¯Ω×R, (i2) the functions aij(x,t) and a(x,t) are continuously differentiable on ¯Ω×R, and bi(x,t,η) are continuously differentiable on ¯Ω×R×Rn, and a(x,t)⩾0 on ¯Ω×R. The nonlinearity g(x,t) satisfies the conditions: (g1) g(x,t) is a Borel function (mod0) on Ω×R [2], that is, it is different from some Borel (on Ω×R) function only on a set ω⊂Ω×R whose projection onto Ω is a nullset; (g2) there exist a nondecreasing (on R+) positive function ψ(t) and a function d(x) from the space Lp(Ω) (p>n) such that, for almost all x∈Ω, (g3) some function β(x) is positive on Ω and satisfies g(x,t)⩾β(x) on Ω×R. Note that condition (g1) implies that g(x,t) is superpositionally measurable on Ω×R [2], that is, for any function u(x) measurable on Ω, the composition g(x,u(x)) is measurable on Ω. If g(x,u) is superpositionally measurable on Ω×R and monotone with respect to u on R, then it is a Borel function (mod0) (see [2]). A Carathéodory function g(x,u) on Ω×R (measurable in x for any u∈R and continuous in u for almost all x∈Ω) is also a Borel function (mod0) (see [2]), even though it may fail to be a Borel function on Ω×R. In what follows, we will also need the following condition: (g4) there exists a Lebesgue nullset ω⊂Ω such that the set is a nullset, and a(x,t)≡0 on ¯Ω×R.Here and in what follows, we use the following notation: for any function g\colon \Omega\times\mathbb R\to\mathbb R such that the section g(x,{\cdot}\,) is a locally bounded function for almost all x\in\Omega.The problem is to study the structure of the set of solutions to problem (1.1), (1.2). Here, by a solution of problem (1.1), (1.2) we will mean an ordered pair (\lambda,u), where \lambda\geqslant 0, and u\in W_p^2(\Omega), p>n, satisfies, in a sense, equation (1.1) and has zero trace on the boundary \partial\Omega. The set of solutions of problem (1.1), (1.2) is considered in the direct product \mathbb R and S, where S is a function space continuously embedded into W_p^1(\Omega), and in which the space W_q^2(\Omega) (p\geqslant q >n, q\geqslant 2) is compactly embedded. In the present paper, S=C^{1,\alpha}(\overline\Omega), 0<\alpha<(p-n)/p. Definition 1. A generalized solution of problem (1.1), (1.2) is a pair (\lambda,u) such that, for almost all x\in\Omega, the function u(x) satisfies the inclusion Lu(x)\in \lambda[g_-(x,u(x)),\,g_+(x,u(x))] and the boundary condition (1.2). The study of the set of generalized solutions of the class of problems of type (1.1), (1.2) requires a special attention because this class involves the well-known Kuiper problem on heating a conductor in a homogeneous electric field of intensity \sqrt\lambda (see [3]). In the Kuiper problem, the differential operator in the left-hand side of equation (1.1) has the form
\begin{equation*}
\begin{aligned} \, -\sum_{i=1}^n(k(x,u(x))u_{x_i})_{x_i} &= -\sum_{i=1}^nk(x,u(x))u_{x_ix_i} \\ &\qquad- \sum_{i=1}^n\bigl(k_{x_i}(x,u(x))+k_u(x,u(x))u_{x_i}\bigr)u_{x_i}. \end{aligned}
\end{equation*}
\notag
Here u(x) is the temperature of the conductor at the point x\in\Omega, k(x,u(x)) is the thermal conductivity. In the Kuiper problem, the role of the nonlinearity g(x,u) is played by the specific electric conductivity, which, under certain temperatures, may have jumps, for example, during recrystallization. In the present paper, the topological method is applied to show that the set U of generalized solutions of problem (1.1), (1.2) in the Banach space \mathbb R\times S contains a connected closed unbounded component containing (0,0). The projection of this component onto \mathbb R is a connected set, hence it is either a half-open interval (possibly, [0,\infty)), or a closed interval with left endpoint 0. For the Kuiper problem, this means that if the squared intensity of the electric field containing the conductor lies in this projection, then there exist a stationary distribution of the temperature. Note that, under the above assumptions, by the maximum principle, for each generalized solution (\lambda,u) with \lambda>0 of problem (1.1), (1.2), the function u(x) is positive almost everywhere on \Omega. Any connected closed unbounded component of the set U of generalized solutions containing (0,0) in the space \mathbb R\times S is called a continuum of generalized positive solutions connecting (0,0) and \infty in the space \mathbb R\times S. Let us recall the definitions of a strong and a semiregular solution of problem (1.1), (1.2). Definition 2. A strong solution of problem (1.1), (1.2) is, by definition, a pair (\lambda,u), where the function u(x)\in W_p^2(\Omega)\cap\mathring{W}^1_p(\Omega) (p>n) satisfies equation (1.1) almost everywhere on \Omega. Definition 3. A semiregular solution of problem (1.1), (1.2) is, by definition, a strong solution (\lambda,u) of this problem such that the set x\in\Omega for which u(x) is a point of discontinuity of the function g(x,{\cdot}\,) is a nullset. If, for almost all x\in\Omega, the inclusion
\begin{equation}
g(x,t)\in [g_-(x,t),\,g_+(x,t)]\quad \forall\, t\in\mathbb R,
\end{equation}
\tag{1.3}
holds, then any strong solution is also a generalized solution. Any semiregular solution of problem (1.1), (1.2) is also a generalized solution of this problem. The concept of a semiregular solution was introduced by Krasnosel’skii and Pokrovskiĭ in [4]. We also mention the papers by Pavlenko and Potapov [5]–[16], which were concerned with the problem of existence of generalized, strong, or semiregular solutions of problem (1.1), (1.2) with a differential operator L linearly depending on u. However, a different machinery for dealing with problem (1.1), (1.2) is required for the quasilinear case considered in the present paper. In [17], Kuiper considered problem (1.1), (1.2) with a differential operator in divergence form, and a(x,u)\leqslant 0 on \Omega\times\mathbb R. In [17], the nonlinearity g(x,u) was assumed to satisfy the following conditions: (a1) g(x,u)=g_0(x,u)+\psi_1(x,u)-\psi_2(x,u), where g_0(x,u) is a Carathéodory function on \Omega\times\mathbb R, the functions \psi_j(x,u) (j=1,2) are continuous in x on \Omega for each u\in\mathbb R and nondecreasing in u on \mathbb R for all x\in\Omega; (a2) the function g_0(x,u) from condition (a1) satisfies the estimate
\begin{equation*}
|g_0(x,u)|\leqslant d(x)+\psi(|u|)\quad\forall\, (x,u)\in\Omega\times\mathbb R,
\end{equation*}
\notag
where d(x)\in L_p(\Omega) (p>n), \psi(t) is a nondecreasing function on \mathbb R_+; (a3) some function \beta(x) positive on \Omega satisfies
\begin{equation*}
g(x,t)\geqslant\beta(x)\quad\forall\, (x,t)\in\Omega\times\mathbb R.
\end{equation*}
\notag
From condition (a1) it follows that the set of points of discontinuity of g(x,u) in the phase variable u is at most countable, that is, the set
\begin{equation*}
D=\{t\in\mathbb R\colon \text {there exists } x\in\Omega \text { such that } g_-(x,t)\neq g_+(x,t)\}
\end{equation*}
\notag
is at most countable (see [17]). If, in condition (a1), the continuity of \psi_j(x,t) in x, is replaced by the condition that it is measurable in x on \Omega, then the set D can be uncountable. Example 1. Let the function f\colon [0,1]\times\mathbb R\to\mathbb R be defined by
\begin{equation*}
f(x,t)=\begin{cases} 2 &\text{if }x<t, \\ 1 &\text{if }x\geqslant t. \end{cases}
\end{equation*}
\notag
This function f is measurable in x and nondecreasing in t. The set D of points of discontinuity of f(x,t) in the phase variable t is [0,1] (this set is uncountable). Since, for any x\in [0,1], the function f(x,{\cdot}\,) is left-continuous on \mathbb R, it is superpositionally measurable (see [18]). Being monotone in t, this function is a Borel function (\operatorname{mod} 0) on [0,1]\times\mathbb R (see [2]). Note that the function f(x,t), which is measurable in x and nondecreasing in t, cannot be superpositionally measurable. Example 2. Let the function f(x,t) by defined on [0,1]\times\mathbb R by
\begin{equation*}
f(x,t)= \begin{cases} 1 &\text{if } x\in [0,1],\, t<1, \\ 2 &\text{if } x\in [0,1],\, t>1, \\ 1 &\text{if } x\in A,\, t=1, \\ 2 &\text{if } x\in [0,1]\setminus A,\, t=1, \end{cases}
\end{equation*}
\notag
where A is a nonmeasurable subset of [0,1]. Then f(x,t) is not superpositionally measurable on [0,1]\times\mathbb R, because the composition f(x,t(x)), where t(x)\equiv 1 on [0,1], is nonmeasurable on [0,1] (this function is equal to 1 for x\in A and 2 for x\in [0,1]\setminus A). Note that in Example 2 the functions f_-(x,t) and f_+(x,t) are superpositionally measurable, because, for each x\in [0,1], the function f_-(x,{\cdot}\,) (f_+(x,{\cdot}\,)) is left (right) continuous on \mathbb R, and, for any u\in\mathbb R, the function f_-(x,u) (f_+(x,u)) is measurable in x on \Omega [18]. In [17], it was shown that problem (1.1), (1.2) admits a continuum of positive solutions in the space \mathbb R\times S connecting (0,0) and \infty. Kuiper’s proof of this result depended on a special approximation of the discontinuous nonlinearity g(x,u) by continuous nonlinearities on \Omega\times\mathbb R. Next, Krasnosel’skii’s theorem (see Theorem 5.5 in [19]) was applied to show that the each approximating problem admits a continuum of positive solutions connecting (0,0) and \infty. Then a passage to the limit was applied to show that the original problem also admits a continuum of positive solutions. Chang [20] studied problem (1.1), (1.2) with an operator L in divergence form via a topological approach under the same assumptions (a1)–(a3) on the right-hand side of equation (1.1) as in Kuiper’s paper. However, unlike Kuiper, Chang assumed that a(x,u) is nonnegative on \Omega\times\mathbb R. Note that the condition a(x,u)\leqslant 0 on \Omega\times\mathbb R was essential for Kuiper, because he additional required that the differential operator L should be coercive (see [17], condition L-2). In [20], Chang showed that problem (1.1), (1.2) has a continuum of positive solutions connecting (0,0) and \infty in \mathbb R\times W_p^1(\Omega), p>n, for more general (than in (1.1)) quasilinear elliptic type operators L in divergence form. According to Chang, in (1.1), (1.2) the existence of a continuum of generalized positive solutions in problem connecting (0,0) and \infty in the space W_p^1(\Omega) follows from Theorem 2.12 in [20]. However, in [20] Chang never verifies the conditions of this theorem. The main results of the present paper are as follows. Theorem 1. Let the following conditions be met: 1) \Omega\subset\mathbb R^n is a C^{1,1}-smooth domain; 2) the coefficients of the differential operator L satisfy conditions (i1), (i2); 3) the nonlinearity g(x,t) satisfies conditions (g1)–(g3) and, for almost all x\in\Omega, inclusion (1.3) holds. Then problem (1.1), (1.2) has a continuum of generalized positive solutions connecting (0,0) and \infty in the space \mathbb R\times C^{1,\alpha}(\overline\Omega), 0<\alpha<(p-n)/p, p>n. Theorem 2. Let the hypotheses of Theorem 1 be met and let condition (g4) hold. Then all generalized solutions of problem (1.1), (1.2) are semiregular. In the proof of Theorem 1, we will first write problem (1.1), (1.2) in the operator form, and then verify the hypotheses of Theorem 2.12 from [20]. We will also give a sufficient condition for semiregularity of generalized solutions of problem (1.1), (1.2) (Theorem 2). In comparison with [17] and [20], the assumptions on the nonlinearity g(x,u) are relaxed: condition (a1) is replaced by the more general condition (g1). In particular, the set of points of discontinuity of g(x,u) in the phase variable u is allowed to be uncountable (see Example 1). § 2. Operator statement of problem (1.1), (1.2). Auxiliary resultsLet p>n and \alpha\in (0,(p-n)/p) be fixed. In this case, the Sobolev space W_p^2(\Omega) is compactly embedded into the Hölder space C^{1,\alpha}(\overline\Omega) (see [21]) with the norm
\begin{equation*}
\begin{aligned} \, \|u\| &=\sup_\Omega|u(x)|+\sup\Bigl\{\sup_\Omega|u_{x_j}|\colon j=1,\dots,n\Bigr\} \\ &\qquad+\sup\biggl\{\sup_{x,y\in\Omega,\, x\neq y} \frac{|u_{x_j}(x)-u_{x_j}(y)|}{|x-y|^\alpha}\colon j=1,\dots,n\biggr\}. \end{aligned}
\end{equation*}
\notag
The norm in the Sobolev space W_q^l(\Omega) (q\geqslant 1, l\in\mathbb N) will be denoted by \|\,{\cdot}\,\|_{q,l}, and the norm in the Lebesgue space L_q(\Omega) (1\leqslant q\leqslant\infty), by \|\,{\cdot}\,\|_q. For each u\in C^{1,\alpha}(\overline\Omega), we define on E=W_p^2(\Omega)\cap\mathring{W}^1_p(\Omega) the differential operator
\begin{equation}
\begin{aligned} \, L(u)v &\equiv-\sum_{i,j=1}^na_{ij}(x,u(x))v_{x_ix_j} \nonumber \\ &\qquad+\sum_{i=1}^nb_i(x,u(x),\nabla u(x))v_{x_i}+a(x,u(x))v\quad \forall\, v\in E \end{aligned}
\end{equation}
\tag{2.1}
with values in L_p(\Omega). Here the functions a_{ij}(x,t), b_i(x,t,\eta) and a(x,t) are the same as in equation (1.1) and satisfy conditions (i1), (i2). Hence the operator L(u) is uniformly elliptic on \Omega, its coefficients are continuous on \overline\Omega, and the coefficient multiplying v is nonnegative on \Omega. In addition, the coefficients a_{ij}(x,u(x)) are continuously differentiable on \overline\Omega. By Lemma 9.17 in [1], there exists a constant M>0 independent of v\in E such that The mapping L(u)\colon E\to L_p(\Omega) is bijective (see [1], Theorem 9.15). Estimate (2.2) implies that the inverse operator L^{-1}(u), u\in C^{1,\alpha}(\overline\Omega), is bounded (E is considered with the norm W_p^2(\Omega)) from L_p(\Omega) into E. In what follows, L^{-1}(u) will be looked upon as an operator from L_p(\Omega) into C^{1,\alpha}(\overline\Omega). Since the embedding of W_p^2(\Omega) into C^{1,\alpha}(\overline\Omega) (p>n, \alpha\in (0,(p-n)/p)) is compact, the operator L^{-1}(u) is compact. The following result holds. Lemma 1. Under conditions (i1), (i2), let B(\theta,R) be the ball in the space C^{1,\alpha}(\overline\Omega) of radius R with centre at the origin \theta of this space. Then there exists a constant M>0 depending only on R such that estimate (2.2) holds with any u\in B(\theta,R). Proof. From conditions (i1), (i2) it follows that, for each u\in B(\theta,R), there exists a constant C>0 independent of v\in E such that
\begin{equation}
\|v\|_{p,2}\leqslant C(\|L(u)v\|_p+\|v\|_p)\quad \forall\, v\in E.
\end{equation}
\tag{2.3}
Here, the constant C depends on \chi(\|u\|_\infty) (\chi(t) is the function from conditions (i1)), \|da_{ij}(x,u(x))/dx_s\|_\infty, \|b_i(x,u(x),\nabla u(x))\|_\infty, \|a(x,u(x))\|_\infty, and \mu(\|u\|_\infty), where \mu(r)=\max\bigl\{\max\{|a_{ij}(x,t)|\colon x\in\overline\Omega,\, |t|\leqslant r\},\ i,j=1,\dots,n\bigr\} [22], p. 199. It can be assumed that C is nondecreasing with respect to each of the above parameters. The quantities \chi(\|u\|_\infty), \|da_{ij}(x,u)/dx_s\|_\infty, \|b_i(x,u,\nabla u)\|_\infty, \|a(x,u)\|_\infty, and \mu(\|u\|_\infty) are bounded on the ball B(\theta,R). Since C depends monotonically on these quantities and since these quantities are uniformly bounded, there exists C=C_1, with which the required estimate holds for all u\in B(\theta,R).
Let us estimate \|v\|_p in terms of \|L(u)v\|_p for all v\in E, and all u\in B(\theta,R). To this end, we will first estimate \sup_\Omega|v(x)|. Writing v(x) as v(x)=v^+(x)+v^-(x), where v^+(x)=\max\{v(x),0\}, v^-(x)=\min\{v(x),0\}, we have (-v(x))^+=-v^-(x), |v(x)|=v^+(x)+(-v)^+(x), \sup_\Omega v^+(x)=\sup_\Omega v(x) and \sup_\Omega (-v)^+(x)=\sup_\Omega (-v(x)). For the operator L(u), let D be the determinant of the matrix A(x,u) with entries a_{ij}(x,u(x)), and let D^*=D^{1/n} be the geometric mean of the eigenvalues of the matrix A(x,u) (by condition (i1), the matrix A is positive definite). Note that D^*\in [\lambda_{\min},\lambda_{\max}], where \lambda_{\min}, \lambda_{\max} are, respectively, the smallest and largest eigenvalues of the matrix A(x,u). Let L(u)v=f, u\in B(\theta,R), v\in E. Then v=L^{-1}(u)f. By the Aleksandrov weak maximum principle (see [1], Theorem 9.1),
\begin{equation*}
\sup_\Omega v(x)\leqslant\sup_{\partial\Omega} v^+(x)+C_2\|f/D^*\|_n,
\end{equation*}
\notag
where C_2 depends monotonically only on n, \operatorname{diam}\Omega, and \|b_j(x,u(x),\nabla u(x))/D^*\|_n, j=1,\dots,n. By condition (i1), there exists a constant \alpha>0 such that \lambda_{\min}>\alpha for all u\in B(\theta,R). Hence, since \|b_j(x,u(x),\nabla u(x))\|_n, j=1,\dots,n, is bounded on B(\theta,R), it can be assumed that the constant C_2/D^* is independent of u\in B(\theta,R). As a result, since v(x)=0 on \partial\Omega, there exists a constant C_3 independent of u\in B(\theta,R) such that \sup_\Omega v^+(x)=\sup_\Omega v(x)\leqslant C_3\|L(u)v\|_n for any v\in E and u\in B(\theta,R). Similarly, replacing v by -v, and taking into account that L(u)(-v)=-f, we arrive at the estimate \sup_\Omega (-v^-(x))=\sup_\Omega (-v(x))\leqslant C_4\|L(u)v\|_n for any v\in E and u\in B(\theta,R). Here the constant C_4 does not depend on u\in B(\theta,R). Therefore,
\begin{equation}
\begin{aligned} \, \sup_\Omega |v(x)| &=\sup_\Omega (v^+(x)+(-v^-(x))) \nonumber \\ &\leqslant \sup_\Omega v^+(x)+\sup_\Omega (-v^-(x)) \leqslant (C_3+C_4)\|L(u)v\|_n \end{aligned}
\end{equation}
\tag{2.4}
for any v\in E and u\in B(\theta,R). Since p>n, the space L_p(\Omega) is continuously embedded into L_n(\Omega). The space L_\infty(\Omega) is continuously embedded into L_p(\Omega). Hence from (2.4) there exists a constant C_5 independent of u\in B(\theta,R) such that \|v\|_p\leqslant C_5 \|L(u)v\|_p for any v\in E and arbitrary u\in B(\theta,R). The last estimate together with inequality (2.3) (with C replaced by C_1) implies (2.2) for arbitrary u\in B(\theta,R) with the constant M=C_1(1+C_5) independent of u\in B(\theta,R). Lemma 1 is proved. Let a function g(x,t) satisfy condition 3) of Theorem 1. With this function, we associate the set-valued mapping G from C^{1,\alpha}(\overline\Omega) into L_p(\Omega) by setting, for any u\in C^{1,\alpha}(\overline\Omega),
\begin{equation}
\begin{aligned} \, G(u) &=\{z\colon \Omega\to\mathbb R\colon z \text { is measurable on } \Omega, \text{ and } \nonumber \\ &\qquad z(x)\in [g_-(x,u(x)),\,g_+(x,u(x))] \text { almost everywhere on } \Omega\}. \end{aligned}
\end{equation}
\tag{2.5}
We will require the following two facts. Proposition 1 (see [23], the lemma). Let T\colon E_1\to E_2 be a locally bounded mapping from a Banach space E_1 to a reflexive space E_2. Then the convexification
\begin{equation*}
T^\Box u:=\bigcap_{\varepsilon>0}\operatorname{\overline{co}}\{y=Tv\colon \|v-u\|_{E_1}<\varepsilon\}
\end{equation*}
\notag
of the operator T is weak-norm closed, that is, y\in T^\Box u whenever u_n\to u, y_n\in T^\Box u_n, y_n\rightharpoonup y (\operatorname{\overline{co}}V is the closed convex hull of the set V in E_2, \rightharpoonup denotes weak convergence). Proposition 2 (see [2], Theorem 27.1). Let f(x,t) be a Borel function (\operatorname{mod} 0) on \Omega\times\mathbb R (\Omega be a bounded domain in \mathbb R^n), and let, for almost all x\in\Omega,
\begin{equation*}
|f(x,t)|\leqslant a(x)+b|t|^{q/s}\quad \forall\, t\in\mathbb R,
\end{equation*}
\notag
where a(x)\in L_s(\Omega), b is a positive constant, s\geqslant 1, q\geqslant 1. The mapping F(u)=f(x,u(x)) is considered as a mapping from L_q(\Omega) into L_s(\Omega). Given a function f(x,{\cdot}\,)\colon \mathbb R\to\mathbb R (x\in\Omega), by f_t^\Box we denote its convexification,
\begin{equation*}
\begin{aligned} \, F_t^\Box(u) &=\{z\colon \Omega\to\mathbb R\colon z(x) \textit { is measurable on } \Omega, \textit { and} \\ &\qquad\qquad z(x)\in f_t^\Box(x,u(x))\textit { almost everywhere on } \Omega \}. \end{aligned}
\end{equation*}
\notag
Then ranges of F_t^\Box and F^\Box lie in L_s(\Omega), and F_t^\Box=F^\Box on L_q(\Omega). The following result holds. Lemma 2. Let a function g(x,t) satisfy condition 3) of Theorem 1, and G be the set-valued mapping defined on C^{1,\alpha}(\overline\Omega) by (2.5). Then 1) the range of G lies in L_p(\Omega), and G is bounded from C^{1,\alpha}(\overline\Omega) into L_p(\Omega); 2) G has convex closed values in L_p(\Omega); 3) G is weak-norm closed, that is, y\in G(u) whenever u_n\to u in C^{1,\alpha}(\overline\Omega), y_n\in G(u_n) and y_n\rightharpoonup y. Proof. By condition (g2), we have G(u)\subset L_p(\Omega) for any u\in C^{1,\alpha}(\overline\Omega), and if U is a bounded set in C^{1,\alpha}(\overline\Omega), then G(U):=\bigcup_{u\in U}G(u) is a bounded set in L_p(\Omega). By condition 3) of Theorem 1, g(x,u(x))\in G(u) for any u\in C^{1,\alpha}(\overline\Omega). This proves assertion 1).
Let us verify 2). Given u\in C^{1,\alpha}(\overline\Omega), y_1, y_2\in G(u), we have for almost all x\in\Omega, j=1,2. Hence, for any t\in [0,1],
\begin{equation*}
\begin{gathered} \, (1-t)y_1(x)+ty_2(x)\geqslant (1-t)g_-(x,u(x))+tg_-(x,u(x))=g_-(x,u(x)), \\ (1-t)y_1(x)+ty_2(x)\leqslant (1-t)g_+(x,u(x))+tg_+(x,u(x))=g_+(x,u(x)) \end{gathered}
\end{equation*}
\notag
almost everywhere on \Omega, that is, for any t\in [0,1] This shows that G(u) is convex.
Now let (y_n)\subset G(u) and y_n\to y in L_p(\Omega). There exists a subsequence (y_{n_k}) of (y_n) such that y_{n_k}(x)\to y(x) almost everywhere on \Omega. We have y_{n_k}\in G(u), and hence g_-(x,u(x))\leqslant y_{n_k}(x)\leqslant g_+(x,u(x)) almost everywhere on \Omega. Making k\to\infty, this gives g_-(x,u(x))\leqslant y(x)\leqslant g_+(x,u(x)) almost everywhere on \Omega, that is, y\in G(u). This verifies that G(u) is closed. Let us now proceed with the proof of assertion 3) in Lemma 2. Let (u_n)\subset C^{1,\alpha}(\overline\Omega), u_n\to u in C^{1,\alpha}(\overline\Omega), y_n\in G(u_n), and y_n\rightharpoonup y in L_p(\Omega). Note that g_t^\Box(x,t)=[g_-(x,t),g_+(x,t)]. Hence G(v)=G_t^\Box(v) for any v\in C^{1,\alpha}(\overline\Omega). Since u_n\to u in C^{1,\alpha}(\overline\Omega), there exists a constant c>0 such that |u_n(x)|\leqslant c for any x\in\overline\Omega and any n\in\mathbb N, and |u(x)|\leqslant c for any x\in\overline\Omega. Let the function \widehat{g}(x,t) on \Omega\times\mathbb R be defined by \widehat{g}(x,t)=g(x,t) for (x,t)\in\Omega\times [-c-\varepsilon,c+\varepsilon] (\varepsilon>0 is fixed), \widehat{g}(x,t)=g(x,-c-\varepsilon) if x\in\Omega, t<-c-\varepsilon, and \widehat{g}(x,t)=g(x,c+\varepsilon) if x\in\Omega, t>c+\varepsilon. This function \widehat{g}(x,t) is a Borel function (\operatorname{mod} 0) on \Omega\times\mathbb R, and further, by condition (g2), we have on \Omega\times\mathbb R, where d(x)\in L_p(\Omega). Therefore, f(x,t)=\widehat{g}(x,t) satisfies the conditions of Proposition 2 with s=p and q\geqslant 1. Applying this result, we conclude that the convexification of the operator F(v)=\widehat{g}(x,v(x)), v\in L_p(\Omega), coincides with F_t^\Box(v) with f(x,t)=\widehat{g}(x,t). By Proposition 1, the mapping F_t^\Box is weak-norm closed qua a set-valued mapping from L_q(\Omega) into L_p(\Omega). The convergence of (u_n) to u(x) in C^{1,\alpha}(\overline\Omega) implies the convergence of (u_n) to u(x) in L_q(\Omega) and since y_n\in G(u_n)=F_t^\Box(u_n), y_n\rightharpoonup y in L_p(\Omega), we have y\in F_t^\Box(u)=G(u). This verifies assertion 3) of Lemma 2, thereby proving the lemma.The following result holds. Lemma 3. Let the coefficients of the differential operator L in equation (1.1) satisfy conditions (i1), (i2), and let p>n, \alpha\in (0,(p-n)/p), u_0\in C^{1,\alpha}(\overline\Omega), and r>0. Then there exists a constant K>0 such that, for any u from the ball B(u_0,r) in the space C^{1,\alpha}(\overline\Omega) and any v\in W_p^2(\Omega), where \|\,{\cdot}\,\| is the norm in C^{1,\alpha}(\overline\Omega). Proof. Let u\in B(u_0,r), v\in W_p^2(\Omega). Let us estimate the function B(x)=|(L(u(x))-L(u_0(x)))v(x)| for any x\in \Omega. We have, for almost all x\in\Omega,
\begin{equation*}
\begin{aligned} \, B(x) &\leqslant\sum_{i,j=1}^n|a_{ij}(x,u(x))-a_{ij}(x,u_0(x))|\cdot|v_{x_ix_j}(x)| \\ &\qquad+\sum_{i=1}^n|b_i(x,u(x),\nabla u(x))-b_i(x,u_0(x),\nabla u_0(x))|\cdot|v_{x_i}(x)| \\ &\qquad+|a(x,u(x))-a(x,u_0(x))|\cdot|v(x)|. \end{aligned}
\end{equation*}
\notag
An application of the Lagrange formula gives, for almost all x\in\Omega,
\begin{equation}
\begin{aligned} \, &B(x) \leqslant\sum_{i,j=1}^n \bigl|a_{{ij}_t}\bigl(x,u_0(x)+\theta_{ij}(u(x)-u_0(x))\bigr)\bigr| \cdot|u(x)-u_0(x)|\cdot|v_{x_ix_j}(x)| \nonumber \\ &\quad + \sum_{i=1}^n\biggl(\sum_{s=1}^n\bigl|b_{i_{\eta_s}}\bigl(x,u_0(x)+\theta_i(u(x)-u_0(x)),\, \nabla u_0(x)+\theta_i\nabla (u-u_0)(x)\bigr)\bigr| \nonumber \\ \nonumber &\qquad \times |u_{x_s}(x)\,{-}\,u_{0_{x_s}}(x)| \\ &\quad {+}\, \bigl|b_{i_t}\bigl(x,u_0(x)\,{+}\,\theta_i(u(x)\,{-}\,u_0(x)),\, \nabla u_0(x)\,{+}\,\theta_i\nabla (u\,{-}\,u_0)(x)\bigr)\bigr| \nonumber \\ &\qquad \times |u(x)-u_0(x)|\biggr)\cdot |v_{x_i}(x)| \nonumber \\ &\quad + \bigl|a_t\bigl(x,u_0(x)+\theta_0(u(x)-u_0(x))\bigr)\bigr|\cdot|u(x)-u_0(x)|\cdot|v(x)|, \end{aligned}
\end{equation}
\tag{2.7}
where \theta_{ij}(x), \theta_i(x), \theta_0(x)\in (0,1). For any u\in B(u_0,r), we have \|u\|\leqslant r+\|u_0\| in the space C^{1,\alpha}(\overline\Omega). By the assumption, the derivatives a_{{ij}_t}(x,t), a_t(x,t) are continuous on \overline\Omega\times\mathbb R, and b_{i_{\eta_s}}(x,t,\eta), b_{i_t}(x,t,\eta) are continuous on \overline\Omega\times\mathbb R\times\mathbb R^n. Hence they are bounded on bounded subsets of \overline\Omega\times\mathbb R, and \overline\Omega\times\mathbb R\times\mathbb R^n, respectively. Consequently, there exists a constant K_1>0 majorizing the values of the derivative of the coefficients of the operator L (see (1.1)) in inequality (2.7) for almost all x\in\Omega and any u\in B(u_0,r). Hence, by inequality (2.7),
\begin{equation*}
\begin{aligned} \, B(x) &\leqslant K_1\|u-u_0\|\sum_{i,j=1}^n|v_{x_ix_j}(x)| \\ &\qquad+K_1\|u-u_0\|\cdot \sum_{i=1}^n\sum_{s=1}^{n+1}1\cdot|v_{x_i}(x)| +K_1\|u-u_0\|\cdot|v(x)| \end{aligned}
\end{equation*}
\notag
almost everywhere on \Omega, for each u\in B(u_0,r) and v\in W_p^2(\Omega). As a result,
\begin{equation*}
\|B\|_p\leqslant K_1(n^2+n(n+1)+1)\cdot\|u-u_0\|\cdot\|v\|_{p,2}
\end{equation*}
\notag
for any u\in B(u_0,r) and v\in W_p^2(\Omega). This therefore establishes (2.6) with the constant K=K_1(2n^2+n+1)>0. Lemma 3 is proved. The existence of a generalized solution of problem (1.1), (1.2) in the space W_p^2(\Omega) (p>n) is equivalent to solvability of the inclusion in space C^{1,\alpha}(\overline\Omega), where the operator L^{-1}(u), u\in C^{1,\alpha}(\overline\Omega) is defined above, and the mapping G is given by (2.5), \lambda\geqslant 0.If we set for any u\in C^{1,\alpha}(\overline\Omega), then inclusion (2.8) assumes the form u\in\lambda\Phi(u), where \lambda\geqslant 0.The following result holds. Lemma 4. Under conditions 1)–3) of Theorem 1, the mapping \Phi, as defined by (2.9), has the following properties: 1) the image of any bounded subset of C^{1,\alpha}(\overline\Omega) under \Phi is precompact in C^{1,\alpha}(\overline\Omega); 2) \Phi has convex compact values in C^{1,\alpha}(\overline\Omega); 3) \Phi is upper semicontinuous on C^{1,\alpha}(\overline\Omega) [24], that is, for any u_0\in C^{1,\alpha}(\overline\Omega) and arbitrary open set U\supset\Phi(u_0), there exists a neighbourhood V of u_0 such that \Phi(V)\subset U. Proof. Let A be a bounded set in C^{1,\alpha}(\overline\Omega), that is, A lies in some ball B(\theta,R) (\theta is the origin of the space, R>0). By Lemma 1 there exists a constant M>0 such that inequality (2.2) holds for any u\in B(\theta,R) and v\in W_p^2(\Omega).
Assertion 1) of Lemma 2 implies that the set G(A) is bounded in L_p(\Omega), that is, there is a constant C>0 such that \|z\|_p\leqslant C for any z\in G(A). This implies the estimate \|L(u)v\|_p\leqslant C for any u\in A and v\in\Phi(u). Hence from estimate (2.2) it follows that \|v\|_{p,2}\leqslant MC for any v\in\Phi(A). Hence the set \Phi(A) is precompact, because W_p^2(\Omega) is compactly embedded in C^{1,\alpha}(\overline\Omega) (p>n, 0<\alpha<(p-n)/p). Let us show that \Phi has convex compact values in C^{1,\alpha}(\overline\Omega). Let u\in C^{1,\alpha}(\overline\Omega). By Lemma 2, G(u) is a bounded convex closed set in L_p(\Omega). For any fixed u, L^{-1}(u) is a compact linear operator from L_p(\Omega) into C^{1,\alpha}(\overline\Omega). Hence the set \Phi(u)=L^{-1}(u)G(u) is convex and precompact. It remains to verify that this set is closed. Let (v_n)\subset\Phi(u) and let v_n\to v in C^{1,\alpha}(\overline\Omega). Then v_n=L^{-1}y_n, where y_n\in G(u). Since L_p(\Omega) is reflexive, and G(u) is bounded, there exists a subsequence (y_{n_k}) weakly converging to y. The bounded set G(u) is convex and closed, and hence y\in G(u). From weak convergence (y_{n_k}) to y and since the operator L^{-1}(u) is compact, it follows that L^{-1}(u)y_{n_k}\to L^{-1}(u)y. Hence v=L^{-1}(u)y\in\Phi(u). This proves that \Phi(u) is closed. Let us now proceed with the proof of assertion 3) of Lemma 4. We need to show that, for any u_0\in C^{1,\alpha}(\overline\Omega) and arbitrary \varepsilon>0, there exists \delta>0 such that \Phi(B(u_0,\delta))\subset\Phi(u_0)+B(\theta,\varepsilon), \theta is the origin of the space C^{1,\alpha}(\overline\Omega). The last inclusion means that, for any u\in B(u_0,\delta) and v\in\Phi(u), there exists v_0\in\Phi(u_0) such that \|v-v_0\|<\varepsilon. Let u_0\in C^{1,\alpha}(\overline\Omega) and \varepsilon>0 be fixed. By Lemma 1, there exists a constant M>0 such that estimate (2.2) holds for each u\in B(u_0,1) and v\in W_p^2(\Omega), The space W_p^2(\Omega) is compactly embedded in C^{1,\alpha}(\overline\Omega), and hence there exists a constant C>0 such that
\begin{equation}
\|v\|\leqslant C\|v\|_{p,2}\quad\forall\, v\in W_p^2(\Omega).
\end{equation}
\tag{2.10}
According to [2], the set-valued mapping G is upper semicontinuous from C(\overline\Omega) into L_p(\Omega). The space C^{1,\alpha}(\overline\Omega) is compactly embedded in C(\overline\Omega), and hence G is upper semicontinuous from C^{1,\alpha}(\overline\Omega) into L_p(\Omega). Hence, from this \varepsilon>0, we can choose \delta_1>0 such that, for any u\in B(u_0,\delta_1),
\begin{equation}
G(u)\subset G(u_0)+B\biggl(\theta,\frac{\varepsilon}{2MC}\biggr),
\end{equation}
\tag{2.11}
where M and C are the constants from inequalities (2.2) and (2.10), respectively. The set \Phi(u_0) is bounded in W_p^2(\Omega), and hence there exists \delta_2>0 such that
\begin{equation}
\delta_2K\|w\|_{p,2}<\frac{\varepsilon}{2MC}\quad\forall\, w\in\Phi(u_0),
\end{equation}
\tag{2.12}
where the constant K corresponds to the ball B(u_0,1) in inequality (2.6) (see Lemma 3).
We put \delta=\min\{1,\delta_1,\delta_2\}. We claim that \Phi(B(u_0,\delta))\subset\Phi(u_0)+B(\theta,\varepsilon). Let u\in B(u_0,\delta) and v\in\Phi(u). Then v=L^{-1}(u)y, where y\in G(u). By (2.11), there exists y_0\in G(u_0) such that \|y-y_0\|_p<\varepsilon/(2MC). We set v_0=L^{-1}(u_0)y_0. Note that v_0\in\Phi(u_0). Let us estimate \|v-v_0\|. Using (2.2) and (2.10), we have
\begin{equation}
\begin{aligned} \, \|v-v_0\| &\leqslant C\|v-v_0\|_{p,2}\leqslant CM\|L(u)(v-v_0)\|_p \nonumber \\ &\leqslant CM\bigl(\|(L(u)-L(u_0))v_0\|_p+\|L(u)v-L(u_0)v_0\|_p\bigr). \end{aligned}
\end{equation}
\tag{2.13}
Note that L(u)v=y, L(u_0)v_0=y_0, and, by Lemma 3,
\begin{equation*}
\|(L(u)-L(u_0))v_0\|_p\leqslant K\|u-u_0\|\cdot\|v_0\|_{p,2}\leqslant K\delta_2\|v_0\|_{p,2}<\frac{\varepsilon}{2MC}
\end{equation*}
\notag
(here, we used inequality (2.12)). Now from (2.13) and in view of the choice of y_0, we have
\begin{equation*}
\|v-v_0\|\leqslant CM\biggl(\frac{\varepsilon}{2MC} +\|y-y_0\|_p\biggr)<\frac{\varepsilon}2+\frac{\varepsilon}2=\varepsilon.
\end{equation*}
\notag
This establishes that \Phi is upper semicontinuous. Lemma 4 is proved. Let X, Y be Banach spaces, S be a metric space, \Gamma(Y) be the set of all nonempty closed convex subsets of Y. Next, let \varphi_1 \colon S\to \Gamma (Y) be a compact mapping, that is, \varphi_1 is upper semicontinuous on S and \varphi_1(S) is a precompact set in Y, and let \varphi_2 \colon Y\to X be a single-valued continuous mapping. In this case, \varphi=\varphi_2\circ\varphi_1\colon S\to 2^X is called a compact sequence of mappings of type I (see [20]). The following result holds. Lemma 5. Let conditions 1)–3) of Theorem 1 be met. Then, for any open bounded subset A of the space C^{1,\alpha}(\overline\Omega), the mapping \Phi, as defined by (2.9), is compact sequence of type I on \overline A. Proof. Let X=Y=C^{1,\alpha}(\overline\Omega), let S=\overline A be the C^{1,\alpha}(\overline\Omega)-closure of A, and let \varphi_1=\Phi\vert {}_{\overline A}, \varphi_2 be the identity mapping on C^{1,\alpha}(\Omega). By Lemma 4, the mapping \varphi_1\colon S\to\Gamma (Y) is compact, and \varphi_2\colon Y\to X is a single-valued continuous mapping. Since \Phi\vert {}_{\overline A}=\varphi_2\circ\varphi_1, it is a compact sequence of type I on \overline A. Lemma 5 is proved. For the proof of the assertion in Theorem 1 on the existence of a continuum of generalized positive solutions connecting (0,0) and \infty, it suffices to verify conditions of Theorem 2.12 from [20]. We recall this result. Theorem 3. Let X, Y be real Banach spaces, where X is semiordered by a cone P, \varphi=\varphi_2\circ\varphi_1\colon [0,R]\times \overline A\to \Gamma(Y)\to 2^P be a compact sequence mappings of type I for each R>0 and arbitrary open bounded set A\subset P. Suppose that \varphi(0,\theta)=\theta (\theta is the origin of X) and \theta is a unique fixed point of the mapping \varphi(0,{\cdot}\,). We also suppose that there exists \rho>0 such that \nu x\notin\varphi(0,x) for any x\in S_\rho^+=\{y\in P\colon \|y\|=\rho\} and \nu\geqslant 1. Then the set of solutions \Sigma=\{(\lambda,x)\in\mathbb{R}_+\times P\colon x\in \varphi(\lambda,x)\} contains an unbounded connected closed subset containing (0,\theta) (that is, a continuum of positive solutions of the inclusion x\in\varphi(\lambda,x) connecting (0,\theta) and \infty). Theorem 3 can be looked upon as an analog of the Leray–Schauder theorem for set-valued mappings. Theorem 3 will be proved by the machinery of degree theory constructed by Chang in [20] for compact sequences of mappings. Below, we will explain what was an obstacle in proving Theorem 1 by the Kuiper scheme. In the first stage of the main theorem in [17], Kuiper establishes the existence of a continuum of positive solutions for the approximating problem with continuous nonlinearity in the right-hand side. The proof of this results depends on a general theorem (see [25], Theorem 2.5), which is a corollary of the Krasnosel’skii proper vector theorem (see [19], Theorem 5.5). Theorem 2.5 in [25] establishes the following result. Let T\colon [0,\infty)\times K\to K be a compact continuous in a Banach space Y semiordered by the cone K. Then there exists a continuum of positive solutions of the equation x=\lambda T(\lambda,x) connecting (0,0) and \infty. Positivity of the solution (\lambda,x) means that \lambda\geqslant 0 and x\in K. If in Theorem 1 the nonlinearity g(x,u) is approximated by continuous nonlinearities (as was done by Kuiper), then by employing Lemma 4 and assumption (g3) one can verify that the operator T_k(u)\equiv L^{-1}(u)G_k(u) on C^{1,\alpha}(\overline\Omega), is compact and continuous, and T_k(K)\subset K, where K is the cone of nonnegative functions in the space C^{1,\alpha}(\overline\Omega). Here G_k is the Nemytskii operator generalized by the nonlinearity g_k(x,u), which approximates g(x,u). According to Theorem 2.5 in [25], there exists a continuum of positive solutions in C^{1,\alpha}(\overline\Omega) of the approximating problems (this also follows from Theorem 17 in [3]). So, the fist part of the proof in the Kuiper scheme is also valid under the hypotheses of Theorem 1. The second stage of Kuiper’s proof is based on a passage to the limit. In his notation, the key Kuiper’s result here is as follows. If a sequence of solutions of the approximating problems converges in C^{1,\alpha}(\overline\Omega), then the limit function is a strong solution of the original problem with discontinuous nonlinearity. In the proof of this result, Kuiper used some assumptions which are not secured by the hypotheses of Theorem 1. Among these assumptions, we mention: assumption (a1) on the structure of the nonlinearity g(x,u) (which implies that the number of points of discontinuity of g(x,u) with respect to u is at most countable), the divergence form of the equation, coercivity of the left-hand side in \mathring{W}^1_2(\Omega) (see [17], condition L-2), and nonpositivity of a(x,u). This was an obstacle in the implementation of the second stage in the Kuiper scheme under the conditions of Theorem 1. The following lemma will be required in the proof of semiregularity of the generalized solutions in Theorem 2. Lemma 6. Let \Omega be a bounded C^{1,1}-smooth domain in \mathbb R^n, let
\begin{equation*}
Lu(x)\equiv-\sum_{i,j=1}^na_{ij}(x)u_{x_ix_j}+\sum_{i=1}^nb_i(x)u_{x_i}
\end{equation*}
\notag
be a uniformly elliptic differential operator in \Omega in which a_{ij}(x) are continuously differentiable and b_i(x) are continuous in \overline\Omega. Next, let u\in W_p^2(\Omega) (p>n) be a strong solution of the problem
\begin{equation*}
Lu(x)=f(x),\quad x\in\Omega; \qquad u(x)=0,\quad x\in\partial\Omega,
\end{equation*}
\notag
where the function f is positive on \Omega and lies in L_p(\Omega), and \sigma\subset\mathbb R is a nullset. Then u^{-1}(\sigma):=\{x\in\Omega\colon u(x)\in\sigma\} is a nullset. Proof. Assume on the contrary that \operatorname{mes}_nu^{-1}(\sigma)\neq 0 (\operatorname{mes}_n is the Lebesgue measure in \mathbb R^n). Since p>n, we have u\in C^1(\overline\Omega). Hence \nabla u(x) is continuous on \overline\Omega. We have u\in W_p^2(\Omega), and hence u_{x_i}(x)\in W_p^1(\Omega), i=1,\dots,n. As a result, if A=\{x\in\overline\Omega\colon \nabla u(x)=0\}, then the generalized derivatives u_{x_ix_j}(x) vanish almost everywhere on A (see [1], Lemma 7.7). As a result, Lu(x)=0 almost everywhere on A. However, this is possible only if \operatorname{mes}_nA=0, inasmuch as Lu(x)=f(x)>0 almost everywhere on \Omega. Since \nabla u(x) is continuous on \overline\Omega, the set A is closed, and hence its complement A^c=\Omega\setminus A is open. The set u^{-1}(\sigma) is measurable and is contained in \Omega. By the assumption, its measure is nonzero. Since \operatorname{mes}_nA=0 and since A^c is open, there exists an n-dimensional parallelepiped \Pi=[a_1,b_1]\times\dots\times[a_n,b_n]\subset A^c such that \operatorname{mes}_n(u^{-1}(\sigma)\cap\Pi)\neq 0. Moreover, \Pi can be chosen so that we have, in addition, |u_{x_{i_0}}(x)|\geqslant\varepsilon on \Pi for some i_0\in \{1,\dots,n\} and \varepsilon>0. It can be assumed without loss of generality that i_0=1. Consider the mapping R\colon \Pi\to\mathbb R^n defined by R(x)=R(x_1,x_2,\dots,x_n)=(u(x),x_2,\dots,x_n) for arbitrary x\in\Pi. The function R\in C^1(\Pi) and its Jacobian are all equal to u_{x_1}(x). By the generalized Sard lemma [26], the mapping R^{-1} sends nullsets to nullsets. Hence R maps sets of positive measure to sets of positive measure. The same property is shared by the function \varrho\colon \mathbb R^n\to\mathbb R, \varrho(y_1,\dots,y_n)=y_1, and hence, by the composition \varrho\circ R, which is equal to u|_\Pi. We have \operatorname{mes}_n(u^{-1}(\sigma) \cap \Pi)>0, and hence by the above the set u(u^{-1}(\sigma)\cap\Pi)\subset\sigma has positive measure. On the other hand, \operatorname{mes}_1\sigma=0 by the hypotheses of Lemma 6. This contradiction proves Lemma 6. Note that the proof of Lemma 6 is similar to that of Theorem 6 in [3]. However, in [3] the operator L is given in divergence form and is supposed to be coercive; the latter condition means that there exists a constant c>0 such that
\begin{equation*}
\int_\Omega\biggl(\sum_{i,j=1}^na_{ij}(x)v_{x_i}v_{x_j}+\sum_{i=1}^nb_i(x)v_{x_i}v\biggr)\, dx\geqslant c\|v\|_{2,1}^2\quad \forall\, v\in\mathring{W}^1_2(\Omega).
\end{equation*}
\notag
Here, Lv\equiv -\sum_{i,j=1}^n(a_{ij}(x)v_{x_i})_{x_j}+\sum_{i=1}^nb_i(x)v_{x_i}.
§ 3. Proof of the main results Proof of Theorem 1. The existence of a generalized solution of problem (1.1), (1.2) in the space W_p^2(\Omega) (p>n) is equivalent to that of the inclusion u\in\lambda\Phi(u) in the space C^{1,\alpha}(\overline\Omega), 0<\alpha<(p-n)/p, \Phi(u)=L^{-1}(u)G(u), where the operator L(u) is defined by (2.1), L^{-1}(u) is the inverse operator of L(u) considered as a self-adjoint operator from L_p(\Omega) into C^{1,\alpha}(\overline\Omega), and the mapping G is given by (2.5), \lambda\geqslant 0.
In the real Banach space X=C^{1,\alpha}(\overline\Omega), we introduce the partial ordering defined by the cone P of nonnegative functions from C^{1,\alpha}(\overline\Omega) as follows: u\leqslant v if u(x)\leqslant v(x) on \overline\Omega. By condition (g3), for some function \beta(x) positive on \Omega, we have g(x,t)\geqslant \beta(x) for any (x,t)\in\Omega\times\mathbb R. Hence by the strong maximum principle [1] the generalized solutions of problem (1.1), (1.2) are positive in \Omega if \lambda>0. This implies that the range of the mapping \varphi(\lambda,u)=\lambda\Phi(u), \lambda\geqslant 0, lies in the cone P. We have \varphi(0,u)\equiv\theta (\theta is the zero element in X), and hence \varphi(0,{\cdot}\,) has a unique fixed point \theta, and moreover, \nu u\notin\varphi(0,u) for any u\neq\theta and arbitrary \nu\geqslant 1. By Lemma 5, for any open bounded set A in the space X, the mapping \Phi is a compact type I sequence on \overline A. It follows that, for any open bounded set A in X and arbitrary R>0, the mapping \varphi(\lambda,u) on [0,R]\times\overline A is a compact type I sequence. So, all the hypotheses of Theorem 3 with Y=X are met. Hence there exists a continuum of generalized positive solutions of problem (1.1), (1.2) which connect (0,\theta) and \infty. Theorem 1 is proved. Proof of Theorem 2. Let (\lambda,u) be a generalized solution of problem (1.1), (1.2). Then there exists a measurable on \Omega function for almost all x\in\Omega such that almost everywhere on \Omega, u(x)=0 on \partial\Omega. From condition (g3) it follows that z(x)> 0 almost everywhere on \Omega. If in equation (1.1) the function a(x,t) is identically zero on \overline\Omega\times\mathbb R, and the functions a_{ij}(x,t), b_i(x,t,\eta) obey conditions (i1), (i2), then the operator L(u(x)) in equation (3.1) satisfies the hypotheses of Lemma 6. From condition (g4) it follows that, except the nullsets \omega\subset\Omega, the set \sigma of values of u(x) for which g_-(x,u(x))\neq g_+(x,u(x)) is a nullset. By Lemma 6, u^{-1}(\sigma) is a nullset. This implies that (\lambda,u) is a semiregular solution of problem (1.1), (1.2). Theorem 2 is proved.
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