
One class of quasilinear elliptic type equations with discontinuous nonlinearities
V. N. Pavlenko^{a}, D. K. Potapov^{b} ^{a} Chelyabinsk State University
^{b} Saint Petersburg State University
Abstract:
In a bounded domain $\Omega\subset \mathbb{R}^n$, a class of quasilinear elliptic type boundary problems with parameter and
discontinuous nonlinearity is studied.
This class of problems includes the H. J. Kuiper conductor heating problem in a homogeneous electric field.
The topological method is applied to verify the existence of a continuum of generalized positive solutions
from the Sobolev space $W_p^2(\Omega)$ ($p>n$) connecting $(0,0)$ with $\infty$
in the space $\mathbb R\times C^{1,\alpha}(\overline\Omega)$, $\alpha\in (0,(pn)/p)$. A sufficient condition
for semiregularity of generalized solutions of this problem is given.
The constraints on the discontinuous nonlinearity
are relaxed in comparison with those used by H. J. Kuiper and K. C. Chang.
Keywords:
quasilinear elliptic type equation, parameter, discontinuous nonlinearity, continuum of positive solutions, semiregular solution, topological method.
Received: 18.04.2021 Revised: 07.02.2022
§ 1. Introduction, statement of the problem, and the main results In a given bounded $C^{1,1}$smooth domain $\Omega\subset \mathbb{R}^n$ (see [1]), consider the following boundaryvalue problem with homogeneous boundary Dirichlet condition for a quasilinear elliptic type equation with nonnegative parameter $\lambda$ and discontinuous nonlinearity $g(x,u)$ of the following form:
$$
\begin{equation}
\begin{split} Lu(x)&\equiv \sum_{i,j=1}^na_{ij}(x,u(x))u_{x_ix_j}+\sum_{i=1}^nb_i(x,u(x),\nabla u(x))u_{x_i} \\ &\qquad +a(x,u(x))u=\lambda g(x,u(x)),\qquad x\in\Omega, \end{split}
\end{equation}
\tag{1.1}
$$
$$
\begin{equation}
u(x)=0,\qquad x\in\partial\Omega.
\end{equation}
\tag{1.2}
$$
It is assumed that the coefficients of the differential operator $L$ satisfy the following conditions: (i1) $a_{ij}(x,t)=a_{ji}(x,t)$ on $\overline\Omega\times\mathbb R$, and there exists a nonincreasing positive on $\mathbb R_+$ function $\chi(t)$ such that, for any $(x,t)\in\overline\Omega\times\mathbb R$,
$$
\begin{equation*}
\sum_{i,j=1}^na_{ij}(x,t)\xi_i\xi_j\geqslant\chi(t)\cdot \xi^2\quad\forall\, \xi\in\mathbb R^n;
\end{equation*}
\notag
$$
(i2) the functions $a_{ij}(x,t)$ and $a(x,t)$ are continuously differentiable on $\overline\Omega\times\mathbb R$, and $b_i(x,t,\eta)$ are continuously differentiable on $\overline\Omega\times\mathbb R\times\mathbb R^n$, and $a(x,t)\geqslant 0$ on $\overline\Omega\times\mathbb R$. The nonlinearity $g(x,t)$ satisfies the conditions: (g1) $g(x,t)$ is a Borel function $(\operatorname{mod} 0)$ on $\Omega\times\mathbb R$ [2], that is, it is different from some Borel (on $\Omega\times\mathbb R$) function only on a set $\omega\subset\Omega\times\mathbb R$ whose projection onto $\Omega$ is a nullset; (g2) there exist a nondecreasing (on $\mathbb R_+$) positive function $\psi(t)$ and a function $d(x)$ from the space $L_p(\Omega)$ ($p>n$) such that, for almost all $x\in\Omega$,
$$
\begin{equation*}
g(x,t)\leqslant d(x)+\psi(t) \quad\forall\, t\in\mathbb R;
\end{equation*}
\notag
$$
(g3) some function $\beta(x)$ is positive on $\Omega$ and satisfies $g(x,t)\geqslant\beta(x)$ on $\Omega\times\mathbb R$. Note that condition (g1) implies that $g(x,t)$ is superpositionally measurable on $\Omega\times\mathbb R$ [2], that is, for any function $u(x)$ measurable on $\Omega$, the composition $g(x,u(x))$ is measurable on $\Omega$. If $g(x,u)$ is superpositionally measurable on $\Omega\times\mathbb R$ and monotone with respect to $u$ on $\mathbb R$, then it is a Borel function $(\operatorname{mod} 0)$ (see [2]). A Carathéodory function $g(x,u)$ on $\Omega\times\mathbb R$ (measurable in $x$ for any $u\in\mathbb R$ and continuous in $u$ for almost all $x\in\Omega$) is also a Borel function $(\operatorname{mod} 0)$ (see [2]), even though it may fail to be a Borel function on $\Omega\times\mathbb R$. In what follows, we will also need the following condition: (g4) there exists a Lebesgue nullset $\omega\subset\Omega$ such that the set
$$
\begin{equation*}
D=\bigcup_{x\in\Omega\setminus\omega}\{u\in\mathbb R\colon g_(x,u)\neq g_+(x,u)\}
\end{equation*}
\notag
$$
is a nullset, and $a(x,t)\equiv 0$ on $\overline\Omega\times\mathbb R$. Here and in what follows, we use the following notation:
$$
\begin{equation*}
g_(x,u)=\liminf_{\eta\to u}g(x,\eta), \quad g_+(x,u)=\limsup_{\eta\to u}g(x,\eta)
\end{equation*}
\notag
$$
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, $S=C^{1,\alpha}(\overline\Omega)$, $0<\alpha<(pn)/p$. 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 wellknown 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 lefthand 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 halfopen 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 leftcontinuous 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 righthand 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 $\mathbb R\times C^{1,\alpha}(\overline\Omega)$, $0<\alpha<(pn)/p$, $p>n$. 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 results Let $p>n$ and $\alpha\in (0,(pn)/p)$ 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
$$
\begin{equation*}
\begin{aligned} \, \u\ &=\sup_\Omegau(x)+\sup\Bigl\{\sup_\Omegau_{x_j}\colon j=1,\dots,n\Bigr\} \\ &\qquad+\sup\biggl\{\sup_{x,y\in\Omega,\, x\neq y} \frac{u_{x_j}(x)u_{x_j}(y)}{xy^\alpha}\colon j=1,\dots,n\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
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
$$
\begin{equation}
\v\_{p,2}\leqslant M\L(u)v\_p\quad \forall\, v\in E.
\end{equation}
\tag{2.2}
$$
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 $C^{1,\alpha}(\overline\Omega)$ ($p>n$, $\alpha\in (0,(pn)/p)$) 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 $\sup_\Omegav(x)$. 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 setvalued 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
$$
\begin{equation*}
T^\Box u:=\bigcap_{\varepsilon>0}\operatorname{\overline{co}}\{y=Tv\colon \vu\_{E_1}<\varepsilon\}
\end{equation*}
\notag
$$
of the operator $T$ is weaknorm 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 $x\in\Omega$,
$$
\begin{equation*}
f(x,t)\leqslant a(x)+bt^{q/s}\quad \forall\, t\in\mathbb R,
\end{equation*}
\notag
$$
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 setvalued 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 weaknorm 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
$$
\begin{equation*}
y_j(x)\in [g_(x,u(x)),\,g_+(x,u(x))]
\end{equation*}
\notag
$$
for almost all $x\in\Omega$, $j=1,2$. Hence, for any $t\in [0,1]$,
$$
\begin{equation*}
\begin{gathered} \, (1t)y_1(x)+ty_2(x)\geqslant (1t)g_(x,u(x))+tg_(x,u(x))=g_(x,u(x)), \\ (1t)y_1(x)+ty_2(x)\leqslant (1t)g_+(x,u(x))+tg_+(x,u(x))=g_+(x,u(x)) \end{gathered}
\end{equation*}
\notag
$$
almost everywhere on $\Omega$, that is, for any $t\in [0,1]$
$$
\begin{equation*}
(1t)y_1+ty_2\in G(u).
\end{equation*}
\notag
$$
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
$$
\begin{equation*}
\widehat{g}(x,t)\leqslant d(x)+\psi(c+\varepsilon)
\end{equation*}
\notag
$$
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 weaknorm closed qua a setvalued 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 $p>n$, $\alpha\in (0,(pn)/p)$, $u_0\in C^{1,\alpha}(\overline\Omega)$, 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 $v\in W_p^2(\Omega)$,
$$
\begin{equation}
\(L(u)L(u_0))v\_p\leqslant K\uu_0\\cdot\v\_{p,2},
\end{equation}
\tag{2.6}
$$
where $\\,{\cdot}\,\$ is the norm in $C^{1,\alpha}(\overline\Omega)$. Proof. Let $u\in B(u_0,r)$, $v\in W_p^2(\Omega)$. Let us estimate the function $B(x)=(L(u(x))L(u_0(x)))v(x)$ for any $x\in \Omega$. We have, for almost all $x\in\Omega$,
$$
\begin{equation*}
\begin{aligned} \, B(x) &\leqslant\sum_{i,j=1}^na_{ij}(x,u(x))a_{ij}(x,u_0(x))\cdotv_{x_ix_j}(x) \\ &\qquad+\sum_{i=1}^nb_i(x,u(x),\nabla u(x))b_i(x,u_0(x),\nabla u_0(x))\cdotv_{x_i}(x) \\ &\qquad+a(x,u(x))a(x,u_0(x))\cdotv(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 \bigla_{{ij}_t}\bigl(x,u_0(x)+\theta_{ij}(u(x)u_0(x))\bigr)\bigr \cdotu(x)u_0(x)\cdotv_{x_ix_j}(x) \nonumber \\ &\quad + \sum_{i=1}^n\biggl(\sum_{s=1}^n\biglb_{i_{\eta_s}}\bigl(x,u_0(x)+\theta_i(u(x)u_0(x)),\, \nabla u_0(x)+\theta_i\nabla (uu_0)(x)\bigr)\bigr \nonumber \\ \nonumber &\qquad \times u_{x_s}(x)\,{}\,u_{0_{x_s}}(x) \\ &\quad {+}\, \biglb_{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 + \bigla_t\bigl(x,u_0(x)+\theta_0(u(x)u_0(x))\bigr)\bigr\cdotu(x)u_0(x)\cdotv(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\uu_0\\sum_{i,j=1}^nv_{x_ix_j}(x) \\ &\qquad+K_1\uu_0\\cdot \sum_{i=1}^n\sum_{s=1}^{n+1}1\cdotv_{x_i}(x) +K_1\uu_0\\cdotv(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\uu_0\\cdot\v\_{p,2}
\end{equation*}
\notag
$$
for any $u\in B(u_0,r)$ and $v\in W_p^2(\Omega)$. This therefore establishes (2.6) with the constant $K=K_1(2n^2+n+1)>0$. Lemma 3 is proved. The existence of a generalized solution of problem (1.1), (1.2) in the space $W_p^2(\Omega)$ ($p>n$) is equivalent to solvability of the inclusion
$$
\begin{equation}
u\in \lambda L^{1}(u)G(u)
\end{equation}
\tag{2.8}
$$
in space $C^{1,\alpha}(\overline\Omega)$, where the operator $L^{1}(u)$, $u\in C^{1,\alpha}(\overline\Omega)$ is defined above, and the mapping $G$ is given by (2.5), $\lambda\geqslant 0$. If we set
$$
\begin{equation}
\Phi(u)=L^{1}(u)G(u)
\end{equation}
\tag{2.9}
$$
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<(pn)/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 $\vv_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)$,
$$
\begin{equation*}
\v\_{p,2}\leqslant M\L(u)v\_p.
\end{equation*}
\notag
$$
The space $W_p^2(\Omega)$ is compactly embedded in $C^{1,\alpha}(\overline\Omega)$, and hence there exists a constant $C>0$ such that
$$
\begin{equation}
\v\\leqslant C\v\_{p,2}\quad\forall\, v\in W_p^2(\Omega).
\end{equation}
\tag{2.10}
$$
According to [ 2], the setvalued mapping $G$ is upper semicontinuous from $C(\overline\Omega)$ into $L_p(\Omega)$. The space $C^{1,\alpha}(\overline\Omega)$ is compactly embedded in $C(\overline\Omega)$, and hence $G$ is upper semicontinuous from $C^{1,\alpha}(\overline\Omega)$ into $L_p(\Omega)$. Hence, from this $\varepsilon>0$, we can choose $\delta_1>0$ such that, for any $u\in B(u_0,\delta_1)$,
$$
\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}
$$
where the constant $K$ corresponds to the ball $B(u_0,1)$ in inequality (2.6) (see Lemma 3).
We put $\delta=\min\{1,\delta_1,\delta_2\}$. We claim that $\Phi(B(u_0,\delta))\subset\Phi(u_0)+B(\theta,\varepsilon)$. Let $u\in B(u_0,\delta)$ and $v\in\Phi(u)$. Then $v=L^{1}(u)y$, where $y\in G(u)$. By (2.11), there exists $y_0\in G(u_0)$ such that $\yy_0\_p<\varepsilon/(2MC)$. We set $v_0=L^{1}(u_0)y_0$. Note that $v_0\in\Phi(u_0)$. Let us estimate $\vv_0\$. Using (2.2) and (2.10), we have
$$
\begin{equation}
\begin{aligned} \, \vv_0\ &\leqslant C\vv_0\_{p,2}\leqslant CM\L(u)(vv_0)\_p \nonumber \\ &\leqslant CM\bigl(\(L(u)L(u_0))v_0\_p+\L(u)vL(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\uu_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*}
\vv_0\\leqslant CM\biggl(\frac{\varepsilon}{2MC} +\yy_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 singlevalued 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 singlevalued 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 setvalued 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 righthand 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 lefthand 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<(pn)/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 selfadjoint 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
$$
\begin{equation*}
z(x)\in [g_(x,u(x)),\,g_+(x,u(x))]
\end{equation*}
\notag
$$
for almost all $x\in\Omega$ such that
$$
\begin{equation}
L(u(x))u(x)=\lambda z(x)
\end{equation}
\tag{3.1}
$$
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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Citation:
V. N. Pavlenko, D. K. Potapov, “One class of quasilinear elliptic type equations with discontinuous nonlinearities”, Izv. RAN. Ser. Mat., 86:6 (2022), 143–160; Izv. Math., 86:6 (2022), 1162–1178
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