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Weak semiregular solutions to the Dirichlet problem for quasilinear elliptic equations in divergence form with discontinuous weak nonlinearities V. N. Pavlenkoa, D. K. Potapovb a Chelyabinsk State University, Chelyabinsk, Russia b St. Petersburg State University, St. Petersburg, Russia
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    § 1. IntroductionIn a bounded domain $\Omega\subset\mathbb R^N$ with boundary $\partial\Omega$ we consider a quasilinear elliptic equation in divergence form
$$
\begin{equation}
-\operatorname{div} (a (|\nabla u|^2) \nabla u)=g(x,u(x)), \qquad x\in\Omega,
\end{equation}
\tag{1.1}
$$
with the Dirichlet boundary condition Here the function $a\colon \mathbb R_+\to \mathbb R_+$ satisfies the following conditions: (a1) the function $a(t)$ is continuous on $\mathbb R_+$, vanishes on $[0,t_0]$ for some $t_0>0$, and there exist positive constants $C_1$, $C_2$ and $p>1$ such that the following inequalities hold:
$$
\begin{equation*}
C_1(t-t_0)^{p-1}\leqslant a(t^2)t\leqslant C_2(t-t_0)^{p-1} \quad\forall\, t\geqslant t_0;
\end{equation*}
\notag
$$
(a2) the function $a(t^2)t$ is nondecreasing on $\mathbb R_+$. As concerns the function $g(x,u)$, the following conditions are assumed to hold: (g1) $g(x,u)$ is superpositionally measurable on $\Omega\times\mathbb R$, that is, the composition $g(x,u(x))$ is measurable on $\Omega$ for any measurable function $u(x)$ on $\Omega$; (g2) for almost all $x\in\Omega$ the section $g(x,\,\cdot\,)$ has finite one-sided limits $g(x,u-)$, $g(x,u+)$ for each $u\in\mathbb R$, $g(x,u-)\leqslant g(x,u+)$, and $g(x,u)\in [g(x,u-),g(x,u+)]$; (g3) the following bound holds for almost all $x\in\Omega$:
$$
\begin{equation*}
|g(x,u)|\leqslant d(x)+k|u|^{s-1} \quad\forall\, u\in\mathbb R,
\end{equation*}
\notag
$$
where $d\in L_{s'}(\Omega)$, $s'=s/(s-1)$, $s\in (1,Np/(N-p))$ for $p<N$ and $s>1$ for $p\geqslant N$, and $k$ is a positive constant. Definition 1. A weak solution to problem (1.1), (1.2) is a function $u\in\mathring{W}^1_p(\Omega)$ such that where $(\,\cdot\,{,}\,\cdot\,)$ denotes the inner product in $\mathbb R^N$. Definition 2. A semiregular solution to problem (1.1), (1.2) is a weak solution $u(x)$ such that where $\operatorname{mes}_N$ is the Lebesgue measure on $\mathbb R^N$. Note that if conditions (a1) and (g1)–(g3) are satisfied, then the integrals in (1.3) exist for all $u$, $v\in\mathring{W}^1_p(\Omega)$. If $g(x,u)\equiv f(x)$, where $f\in W_q^{-1}(\Omega)$, $q=p/(p-1)$, $p>1$, then (1.1) is the equation of nonlinear stationary filtration of an incompressible fluid. In this case $u(x)$ is interpreted as the pressure, $\vec v(x)= -a(|\nabla u|^2) \nabla u(x)$ is the filtration velocity, and $f(x)$ is the density of external sources. In this formulation problem (1.1), (1.2) was studied in [1]. The existence of a weak solution was proved there by the method of monotone operators and the smoothness of the weak solution was investigated using results from [2]. The regularization method was used to solve the problem numerically. In [2] the homogeneous Dirichlet problem for a quasilinear elliptic equation
$$
\begin{equation}
-\operatorname{div}b(\nabla u)=f(x), \qquad x\in\Omega,
\end{equation}
\tag{1.4}
$$
was considered, where $b(y)$ is a continuous vector function on $\mathbb R^N$ with components $b^i(y)$, $i=1,\dots,N$, satisfying the following conditions: (b1) for all $y, z\in\mathbb R^N$ the inequality holds, where $(\,\cdot\,{,}\,\cdot\,)$ is the inner product in $\mathbb R^N$ (the monotonicity condition);(b2) for each $y=(y_1,\dots,y_N)\in\mathbb R^N$
$$
\begin{equation*}
|b^i(y)|\leqslant\mu(1+\sum_{j=1}^N|y_j|^p)^{1/q}, \qquad i=1,\dots,N,
\end{equation*}
\notag
$$
where $\mu$ is a positive constant and $q=p/(p-1)$; (b3) for each $y=(y_1,\dots,y_N)\in\mathbb R^N$ where $\nu>0$ and $\mu_1\geqslant 0$.Under these conditions the Galerkin method was used to prove the existence of a weak solution to (1.4), (1.2) for each $f\in W_q^{-1}(\Omega)$, $q=p/(p-1)$, $p>1$. The smoothness of the weak solution was investigated in the case when $f\in L_q(\Omega) \cap W_q^1(\Omega')$ for every interior subdomain $\Omega'$ of $\Omega$. If we set $b^i(y)=a(|y|^2)y_i$, $i=1,\dots,N$, then conditions (b1)–(b3) are satisfied for the vector function $b(y)$ with components $b^i(y)$ if $a(t)$ satisfies conditions (a1) and (a2). Note some features of problem (1.1), (1.2) under consideration: (1) the right-hand side of (1.1) can be discontinuous with respect to the phase variable $u$; (2) the differential operator on the left-hand side of (1.1) is degenerate, since $a(t)=0$ on $[0,t_0]$ by condition (a1), where $t_0>0$. The main result in this paper is as follows. (1) the function $a(t)$ satisfies conditions (a1) and (a2); (2) the function $g(x,u)$ satisfies conditions (g1)–(g3); (3) for almost all $x\in\Omega$ the inequality
$$
\begin{equation*}
\int_0^u g(x,\tau)\,d\tau\leqslant M|u|^p+r_1(x)|u|^{p-\theta}+r(x) \quad\forall\, u\in\mathbb R
\end{equation*}
\notag
$$
holds, where $M\geqslant 0$, $r_1\in L_{p/\theta}(\Omega)$, $\theta\in (0,p)$ and $r\in L_1(\Omega);$ (4) the inequality holds, where $C_1$ is from condition (a1), $M$ is from condition (3), and $P$ is the operator embedding the space $\mathring{W}^1_p(\Omega)$ in $L_p(\Omega)$.Then problem (1.1), (1.2) has a weak semiregular solution in the Sobolev space $\mathring{W}^1_p(\Omega)$. We note that if $M=0$, then assumption $(4)$ in Theorem 1 is obviously satisfied. Next, we introduce a positive parameter $\lambda$ into (1.1) by multiplying the right-hand side of (1.1) by $\lambda$. The new equation has the form
$$
\begin{equation}
-\operatorname{div} (a (|\nabla u|^2) \nabla u)=\lambda g(x,u(x)).
\end{equation}
\tag{1.5}
$$
Then the following result holds. Theorem 2. Let the assumptions of Theorem 1 be satisfied for $M=0$ in assumption $(3)$. Suppose additionally that $g(x,0)=0$ almost everywhere on $\Omega$ and there exists an element $\widehat{u}\in\mathring{W}^1_p(\Omega)$ for which the following inequality holds: Theorems 1 and 2 are proved by a variational method based on the concept of quasipotential operator (see § 2 below). This approach was previously applied by these authors in [3]–[6] to prove the existence of semiregular solutions of semilinear elliptic boundary value problems and integral equations with discontinuous nonlinearities (including ones with a parameter). In comparison to other authors, in the proof of the existence of strong and semiregular solutions the restrictions on points of discontinuity of nonlinearities were relaxed. The traditional variational method of the study of nonsmooth functionals is based on the concept of Clarke generalized derivative for locally Lipschitz functions. As applied to elliptic boundary value problems with discontinuous nonlinearities, this method was developed in [7]. It can be used to obtain existence theorems for generalized solutions. In our setting, a generalized solution to problem (1.1), (1.2) is a function $u\in\mathring{W}^1_p(\Omega)$ such that there exists a measurable function $\eta(x)$ on $\Omega$ such that $\eta(x)\in [g(x,u(x)-),g(x,u(x)+)]$ almost everywhere on $\Omega$ which satisfies (1.3) for $g(x,u(x))$ replaced by $\eta(x)$. In [7] the classical theory of critical points of smooth functionals was generalized to locally Lipschitz functions. Subsequently, a significant number of publications appeared, whose authors, based on this theory, obtained many new results on elliptic equations with discontinuous nonlinearities (see the monograph [8] and the bibliography there). The theory of critical points was further developed in a series of works (see, for example, [9]). Within the framework of the traditional variational approach, the question of the existence of weak and strong solutions was also considered. In this case, additional conditions were imposed on the discontinuity points of the nonlinearity, and it was established with their help that the generalized solution thus obtained is weak or strong. The most general constraint on discontinuity points of the nonlinearity in those papers was as follows (for instance, see [9] and [10]): (i) there exists a set $\Omega_0\subset\Omega$ of measure zero such that the set
$$
\begin{equation*}
Z_g(\Omega_0)=\bigcup_{x\in\Omega\setminus\Omega_0} \{u\in\mathbb R\colon u \text{ is a discontinuity point of the function } g(x,\,\cdot\,)\}
\end{equation*}
\notag
$$
has measure zero. We note that when the assumptions of Theorem 1 are satisfied, constraint (i) may not hold. Let us give an example. Example 1. Let $\varphi(x)$ be a continuously differentiable, positive nonconstant function on $\overline\Omega$ and $\Omega$ be a bounded domain in $\mathbb R^N$. Consider the function on $\overline\Omega\times\mathbb R$. Then $\Omega_0\subset\Omega$ and the measure of $\Omega_0$ is zero. In this case We claim that $\operatorname{mes}_N Z_g(\Omega_0)\neq 0$. We note first that for a continuously differentiable function $f(x)$ on the interval $[a,b]$, the image of a subset of measure zero of $[a,b]$ has measure zero. This follows from the fact that the measure of the $f$-image of the interval $(c,d)\subset [a,b]$ does not exceed $(d-c)\max_{x\in [a,b]}|f'(x)|$. It is interesting to note that this does not hold for continuous functions on $[a,b]$. For example, the Cantor staircase on the interval $[0,1]$ maps the Cantor set in $[0,1]$ to a set of measure $1$. We claim now that $\operatorname{mes}_N Z_g(\Omega_0)\neq 0$. Since $\varphi(x)$ is not constant on $\Omega$, there exists $x_0\in\Omega$ such that $\operatorname{grad}\varphi(x_0)\neq 0$. Consequently, at least one partial derivative of $\varphi$ at $x_0$ is nonzero. Let it be $\partial\varphi/\partial x_i$, $x=(x_1,\dots,x_N)$. Without loss of generality we can assume that $i=1$. Then there exists an $N$-block $\Pi=[a_1,b_1]\times\dots\times [a_N,b_N]\subset\Omega$ on which $\partial\varphi/\partial x_1\neq 0$. Let $B$ denote the intersection of $\Pi$ with $\Omega_0$, and for each $x'=(x_2,\dots,x_N)\in\Pi'=[a_2,b_2]\times\dots\times [a_N,b_N]$ set Since $B$ is a measurable subset of $\Pi\,{=}\,[a_1,b_1]\,{\times}\,\Pi'$, the section $B(x')$ is a measurable subset of $[a_1,b_1]$ for almost every $x'{\in}\,\Pi'$, and $\displaystyle\operatorname{mes}_N B\,{=}\!\int_{\Pi'} \!\operatorname{mes}_1 B(x')\,dx'$ (see [11], Ch. V, § 6, Theorem 3). Since $\operatorname{mes}_N \Omega_0=0$ and $B\subset\Omega_0$, it follows that $\operatorname{mes}_N B\,{=}\,0$. Therefore, there exists $x'\in\Pi'$ such that $\operatorname{mes}_1 B(x')=0$. Hence, by virtue of what we proved above $\operatorname{mes}_1 \varphi(B(x'))=0$. Since $\partial\varphi/\partial x_1\neq 0$ on the closed interval $l=\{(t,x')\colon t\in [a_1,b_1]\}$, the linear measure of the $\varphi$-image of this closed interval is positive. Consequently, $\operatorname{mes}_1 \varphi(l\setminus B(x'))>0$. This immediately implies that $\operatorname{mes}_1 \varphi(\Pi\setminus\Omega_0)\neq 0$ because $l\setminus B(x')\subset\Pi\setminus\Omega_0$. Since $\Pi\subset\Omega$, the linear measure of the $\varphi$-image of $\Omega\setminus\Omega_0$ is positive, as required.Thus, the constraint (i) fails for the nonlinearity of $g(x,u)$. However, $g(x,u)$ satisfies the assumptions of Theorem 1. Indeed, since $g(\cdot,u)$ is measurable on $\Omega$ for each $u\in\mathbb R$, and $g(x,\,\cdot\,)$ is right-continuous on $\mathbb R$ for arbitrary $x\in\Omega$, the function $g(x,u)$ is superpositionally measurable on $\Omega\times\mathbb R$ (see [12]). Therefore, condition (g1) is satisfied. The function $g(x,u)$ also obviously satisfies conditions (g2) and (g3). By the bound
$$
\begin{equation*}
\int_0^u g(x,\tau)\,d\tau\leqslant 1 \quad \forall\, x\in\Omega,\quad u\in\mathbb R,
\end{equation*}
\notag
$$
assumption (3) of Theorem 1 is also satisfied. Among the recent works where the standard variational method was applied to a quasilinear elliptic equation with discontinuous nonlinearity and the question of the existence of semiregular solutions was investigated, we can mention [13]. In a bounded domain $\Omega\subset\mathbb R^N$, the Dirichlet problem for the quasilinear elliptic equation
$$
\begin{equation}
-\operatorname{div}(\varphi(|\nabla u|)\nabla u)=\lambda f(x,u)\chi_{[u\geqslant a]}, \qquad x\in\Omega,
\end{equation}
\tag{1.7}
$$
was studied, where $\varphi\colon [0,+\infty)\,{\to}\, (0,+\infty)$, $\varphi(0)\,{=}\,0$, is a differentiable function, $\lambda$ and $a$ are positive parameters, $\chi_{[u\geqslant a]}$ is the characteristic function of the set $[u\geqslant a]=\{x\in\Omega\colon u(x)\geqslant a\}$, and the function $f\colon \overline\Omega\times\mathbb R\to\mathbb R$ is continuous. This problem was considered in Orlicz–Sobolev spaces. Note that taking $\varphi(t)=a(t^2)$ in (1.7) we obtain the left-hand side of (1.1). In this case $\varphi(t)=0$ for $t\in [0,\sqrt{t_0}\,]$ (see condition (a1)), which is not consistent with the assumption of the positivity of $\varphi$ on $(0,+\infty)$. In [13] the question of the existence or nonexistence of weak semiregular solutions of problem (1.7), (1.2) such that $\operatorname{mes}\{x\in\Omega\colon u(x)\geqslant a\}\neq 0$ was investigated (the authors called them $S$-solutions). In this case, some conditions additional to those indicated above were imposed on $\varphi$ and $f$. For any $x\in\Omega$ the nonlinearity in the right-hand side of (1.7) can have a discontinuity only for $u=a$. In [13] it was noted that the results on the existence of $S$-solutions are easily extended to the case when the right-hand side of (1.7) has the form
$$
\begin{equation}
\lambda f(x,u)\sum_{i=1}^\infty \chi_{[a_i\leqslant u<a_{i+1}]},
\end{equation}
\tag{1.8}
$$
where $a_1=0$ and $a_{i+1}-a_i\geqslant\delta$ for $i\geqslant 1$ and some $\delta>0$. Note that condition (i) is satisfied in this case, since the set $Z_g(\Omega_0)$ is at most countable. Here $\Omega_0\subset\Omega$ has measure zero and $g(x,u)$ is defined by (1.8). In conclusion, we note the paper [14] devoted to the Dirichlet problem for a quasilinear elliptic equation with a parameter and a discontinuous nonlinearity. The existence of continuum many generalized positive solutions connecting $(0,0)$ and $\infty$ was proved there by a topological method. A sufficient condition for the semiregularity of generalized solutions of the boundary value problem under study was obtained. § 2. PreliminariesLet $E$ be a real Banach space and let $E^*$ be the dual space of $E$. We denote the value of a functional $y\in E^*$ at an element $x\in E$ by $\langle y, x\rangle$. Let us present necessary definitions and auxiliary results. Definition 3. An operator $T\colon E\to E^*$ is said to be (1) radially summable if for all $u,h\in E$ the function is integrable on $[0,1]$;(2) radially continuous at a point $u\in E$ if the following equality holds for each $h\in E$:
$$
\begin{equation}
\lim_{t\to 0} \langle T(u+th), h\rangle=\langle Tu, h\rangle;
\end{equation}
\tag{2.1}
$$
(3) monotone if the inequality holds for all $u$, $v\in E$;(4) bounded (compact) if for each bounded set $B\subset E$ the image $T(B)$ is a bounded (precompact) subset of $E^*$; (5) locally bounded on $E$ if for each $u\in E$ there exist a ball $B(u,r)$ and a positive constant $C$ depending on $B(u,r)$ such that the inequality $\|Tv\|\leqslant C$ holds for all $v\in B(u,r)$. Definition 4. A function $f\colon E\to\mathbb R$ is said to be (1) weakly lower semicontinuous at a point $u\in E$ if for each sequence $(u_n)$ weakly converging to $u$ in $E$ we have $\lim_{n\to\infty}\inf f(u_n)\geqslant f(u)$; (2) locally Lipschitz on $E$ if for each $u \in E$ there exist a ball $B(u,r)$ and a positive constant $C$ depending on $B(u,r)$ such that the inequality $|f(v)-f(w)|\leqslant C\|v-w\|$ holds for all $v$, $w\in B(u,r)$; (3) Gâteaux differentiable at a point $u\in E$ if there exists $y\in E^*$ such that for each $h\in E$ there exists a limit (in this case $y$ is called the Gâteaux derivative of $f$ at the point $u$ and is denoted by $f'(u)$).Definition 5. An operator $T\colon E\to E^*$ is said to be potential if there exists a Gâteaux differentiable functional $f$ on $E$ such that $f'(u)=Tu$ for each $u\in E$. In this case $f$ is called a potential of the operator $T$. Definition 6. A radially summable operator $T\colon E\to E^*$ is said to be quasipotential (see [15], Ch. V, § 17, Definition 17.3) if there exists a functional $f\colon E\to\mathbb R$ for which the following equality holds: for all $u,h\in E$. In this case $f$ is called a quasipotential of the operator $T$. We note that if $T$ is potential and radially continuous, then its potential $f$ satisfies (2.2) since We note also that if an operator $T$ is quasipotential and locally bounded, then its quasipotential is a locally Lipschitz function.Definition 7. A point of discontinuity of an operator $T\colon E\to E^*$ is an element $u\in E$ at which the radial continuity condition (2.1) fails. Definition 8. A regular point of an operator $T\colon E\to E^*$ is an element $u\in E$ for which there exists $h\in E$ such that $\lim_{t\to 0+}\sup\langle T(u+th), h\rangle<0$. The following result holds. Proposition 1 ([16], Proposition 1). If $u$ is a discontinuity point of $T\colon E\to E^*$ and for each $h\in E$ the limit exists, then $u$ is a regular point of $T$. The following assertion holds. Theorem 3 ([16], Theorem 1). Let $T\colon E\to E^*$ be a locally bounded quasipotential operator on $E$ whose discontinuity points are regular, and let $x$ be a minimum point of a quasipotential of $T$. Then $T$ is radially continuous at $x$ and $Tx=0$. In addition to Proposition 1 and Theorem 3, the following result is also used in the proofs of Theorems 1 and 2. Theorem 4 ([17]). Let $T=T_1-T_2$, where the $T_i\colon E\to E^*$ are quasipotential operators with quasipotentials $f_i$, respectively, $i=1,2$, let the operator $T_1$ be monotone and the operator $T_2$ be compact. Then the quasipotential $f(u)=f_1(u)-f_2(u)$ of $T$ is weakly lower semicontinuous on $E$. § 3. Proof of the main results Proof of Theorem 1. Let $E=\mathring{W}^1_p(\Omega)$. This is a reflexive Banach space with the norm The dual space of $E$ is $E^*=W_q^{-1}(\Omega)$, $q=p/(p-1)$. Note that $E$ is compactly embedded in $L_s(\Omega)$, where $s<Np/(N-p)$ if $p<N$ and $s>1$ if $p\geqslant N$ (see condition (g3)).
We proceed with the operator formulation of problem (1.1), (1.2) in the space $E$. We define operators $T_i\colon E\to E^*$, $i=1,2$, by the equalities
$$
\begin{equation*}
\langle T_1u, v\rangle=\int_\Omega a(|\nabla u|^2)(\nabla u,\nabla v)\,dx\quad\text{and} \quad \langle T_2u, v\rangle=\int_\Omega g(x,u(x))v(x)\,dx
\end{equation*}
\notag
$$
for all $u$, $v\in E$. We claim that for fixed $u\in E$ the left-hand sides of these equalities define bounded linear functionals on $E$.
For any $v\in E$ and almost all $x\in\Omega$ we have
$$
\begin{equation}
|a(|\nabla u|^2)(\nabla u,\nabla v)|\leqslant a(|\nabla u|^2)|\nabla u|\,|\nabla v|\leqslant C_2|\nabla u|^{p-1}|\nabla v|
\end{equation}
\tag{3.1}
$$
(we have used property (a1)). Since $|\nabla u|^{p-1}\in L_q(\Omega)$ and $\nabla v\in L_p(\Omega)$, it follows by Hölder’s inequality that $|\langle T_1u, v\rangle|\leqslant C_2\|u\|_{p,1}^{p-1}\|v\|_{p,1}$. This implies that $T_1u\in E^*$ and $\|T_1u\|\leqslant C_2\|u\|_{p,1}^{p-1}$.
By condition (g3) the following inequality holds for any $v\in E$ and almost all ${x\in\Omega}$:
$$
\begin{equation}
|g(x,u(x))v(x)|\leqslant (|d(x)|+k|u(x)|^{s-1})|v(x)|.
\end{equation}
\tag{3.2}
$$
Since $|d(x)|+k|u(x)|^{s-1}\in L_{s'}(\Omega)$, where $s'=s/(s-1)$ and $v\in L_s(\Omega)$ (the restrictions on $s$ in (g3) ensure the compactness of the embedding of $E$ in $L_s(\Omega)$), it follows from Hölder’s inequality that for $v\in L_s(\Omega)$ we have
$$
\begin{equation*}
\begin{aligned} \, |\langle T_2u, v\rangle| &\leqslant \bigl\||d|+k|u|^{s-1}\bigr\|_{s'}\|v\|_s \\ &\leqslant(\|d\|_{s'}+k\bigl\| |u|^{s-1}\bigr\|_{s'})\|v\|_s=(\|d\|_{s'}+k\|u\|_s^{s-1})\|v\|_s, \end{aligned}
\end{equation*}
\notag
$$
where $\|\cdot\|_{\widehat{s}}$ is the norm in the space $L_{\widehat{s}}(\Omega)$. Hence, from the compactness of the embedding of $E$ in $L_s(\Omega)$ we obtain the bound
$$
\begin{equation*}
|\langle T_2u, v\rangle|\leqslant (\|d\|_{s'}+k\|P_1\|^{s-1}\|u\|_{p,1}^{s-1}) \|P_1\| \|v\|_{p,1} \quad \forall\, v\in E,
\end{equation*}
\notag
$$
where $P_1$ is the operator embedding the space $E$ in $L_s(\Omega)$. This implies that $T_2u\in E^*$ and $\|T_2u\|\leqslant \|d\|_{s'}\|P_1\|+k\|P_1\|^s\|u\|_{p,1}^{s-1}$.
The existence of a weak solution to problem (1.1), (1.2) is equivalent to the existence of a classical solution of the equation in the space $E$, that is, to the existence of $u\in E$ satisfying (3.3).Let us prove that $T_1$ and $T_2$ are quasipotential operators. For all $u,h\in E$ we have
$$
\begin{equation*}
\begin{aligned} \, &\int_0^1 \langle T_1(u+th), h\rangle \,dt =\int_0^1 dt\int_\Omega a(|\nabla u+t\nabla h|^2)(\nabla(u+th),\nabla h)\,dx \\ &\qquad =\int_0^1 dt\int_\Omega a(|\nabla u|^2+2t(\nabla u,\nabla h)+t^2|\nabla h|^2)((\nabla u,\nabla h)+t|\nabla h|^2)\,dx. \end{aligned}
\end{equation*}
\notag
$$
We change the order of integration and make the change of variable $\tau=|\nabla u|^2+2t(\nabla u,\nabla h)+t^2|\nabla h|^2$ in the inner integral. Then, taking the equality $d\tau=2(\nabla u,\nabla h)\,dt+2t|\nabla h|^2\,dt$ into account we obtain
$$
\begin{equation}
\int_0^1 \langle T_1(u+th), h\rangle \,dt= \int_\Omega dx\int_{|\nabla u|^2}^{|\nabla u+\nabla h|^2}a(\tau)\,\frac{d\tau}{2}= f_1(u+h)-f_1(u) \quad \forall\, u,h\in E,
\end{equation}
\tag{3.4}
$$
where
$$
\begin{equation*}
f_1(u)=\frac12 \int_\Omega dx\int_0^{|\nabla u|^2}a(\tau)\,d\tau.
\end{equation*}
\notag
$$
Thus, $T_1$ is a quasipotential operator and $f_1$ is a quasipotential of $T_1$.
The fact that $T_2$ is quasipotential is proved similarly. For all $u,h\in E$ we have
$$
\begin{equation*}
\int_0^1 \langle T_2(u+th), h\rangle\, dt=\int_0^1 dt\int_\Omega g(x,u(x)+th(x))h(x)\,dx.
\end{equation*}
\notag
$$
We change the order of integration and make the change $\tau=u(x)+th(x)$ in the inner integral. Then, taking the equality $d\tau=h(x)\,dt$ into account we obtain
$$
\begin{equation*}
\int_0^1 \langle T_2(u+th), h\rangle\, dt= \int_\Omega dx\int_{u}^{u+h}g(x,\tau)\,d\tau=f_2(u+h)-f_2(u) \quad \forall\, u,h\in E,
\end{equation*}
\notag
$$
where Therefore, $T_2$ is a quasipotential operator and $f_2$ is a quasipotential of $T_2$.
Note that the operator $T_1$ is radially continuous on $E$. Indeed, since the function $a(\tau)$ is continuous, it follows that
$$
\begin{equation*}
\lim_{t\to 0}a(|\nabla (u+th)|^2)(\nabla (u+th),\nabla h)=a(|\nabla u|^2)(\nabla u,\nabla h)
\end{equation*}
\notag
$$
almost everywhere on $\Omega$ for any $u,h\in E$. Therefore, taking (3.1) into account we have
$$
\begin{equation*}
\begin{aligned} \, \lim_{t\to 0} \langle T_1(u+th), h\rangle &=\lim_{t\to 0}\int_\Omega a(|\nabla (u+th)|^2)(\nabla(u+th),\nabla h)\,dx \\ &=\int_\Omega a(|\nabla u|^2)(\nabla u,\nabla h)\,dx=\langle T_1u, h\rangle \end{aligned}
\end{equation*}
\notag
$$
(we have used Lebesgue’s dominated convergence theorem).
The radial continuity of the operator $T_1$ and (3.4) immediately imply that $T_1$ is a potential operator and $f_1$ is a potential of $T_1$. For $y,z\in\mathbb R^n$ we have
$$
\begin{equation*}
\begin{aligned} \, (a(|y|^2)y-a(|z|^2)z,y-z) &=a(|y|^2)|y|^2+a(|z|^2)|z|^2-(a(|y|^2)+a(|z|^2))(y,z) \\ &\geqslant a(|y|^2)|y|(|y|-|z|)+a(|z|^2)|z|(|z|-|y|) \\ &=(a(|y|^2)|y|-a(|z|^2)|z|)(|y|-|z|)\geqslant 0. \end{aligned}
\end{equation*}
\notag
$$
Here we have used the inequality $|(y,z)|\leqslant |y|\, |z|$ and condition (a2). The inequality thus obtained is used to prove the monotonicity of the operator $T_1$.
For $u,v\in E$ we have
$$
\begin{equation*}
\langle T_1u-T_1v, u-v\rangle=\int_\Omega (a(|\nabla u|^2)\nabla u-a(|\nabla v|^2)\nabla v,\nabla u-\nabla v)\,dx\geqslant 0,
\end{equation*}
\notag
$$
since, by virtue of the inequality proved above, the integrand is nonnegative almost everywhere on $\Omega$. The last inequality means the monotonicity of $T_1$.
Note that the above proof of the monotonicity of the operator $T_1$ differs from the one in [1] and does not assume the absolute continuity of the function $a(t)$. Consider the Nemytskii operator $Gu\equiv g(x,u(x))$ generated by the function $g(x,u)$. By conditions (g1) and (g3) the operator $G\colon L_s(\Omega)\to L_{s'}(\Omega)$ is bounded. The embedding operator $P_1\colon E\to L_s(\Omega)$ is defined by the equality $P_1u=u$ for all $u\in E$. As noted above, $P_1$ is compact. Therefore, the operator $P_1^*$ adjoint to it is compact. For arbitrary $u$, $v\in E$ we have
$$
\begin{equation*}
\langle T_2u, v\rangle=\int_\Omega g(x,u(x))v(x)\,dx=\langle GP_1u, P_1v\rangle=\langle P_1^*GP_1u, v\rangle.
\end{equation*}
\notag
$$
From this we conclude that $T_2=P_1^*GP_1$, and therefore $T_2$ is compact, since $G$ is bounded and $P_1^*$ is compact.
Let us obtain a lower bound on $E$ for the quasipotential $f(u)=f_1(u)-f_2(u)$ of the operator $T=T_1-T_2$. To do this we obtain first a lower bound for $f_1$ and an upper bound for $f_2$ on $E$. It follows from condition (a1) that for each $t\geqslant 2t_0$ we have
$$
\begin{equation*}
a(t^2)t\geqslant C_1(t-t_0)^{p-1}=C_1t^{p-1}\biggl(1-\frac{t_0}{t}\biggr)^{p-1}\geqslant\frac{C_1}{2^{p-1}}t^{p-1}
\end{equation*}
\notag
$$
(the last inequality is a consequence of $t_0/t\leqslant 1/2$). Let $s=t^2$. Then $a(s)\sqrt s\geqslant({C_1}/{2^{p-1}})s^{(p-1)/2}$ for $s\geqslant 4t_0^2$. This implies that
$$
\begin{equation}
a(s)\geqslant\frac{C_1}{2^{p-1}}s^{p/2-1} \quad\forall\, s\geqslant 4t_0^2.
\end{equation}
\tag{3.5}
$$
We use the bound (3.5) to estimate the functional $f_1$ from below. For arbitrary $u\in E$ we have
$$
\begin{equation}
\begin{aligned} \, \notag f_1(u) &=\frac12\int_\Omega dx\int_0^{|\nabla u|^2}a(s)\,ds \\ \notag &=\frac12\biggl(\int_{\Omega_0} dx\int_0^{|\nabla u|^2}a(s)\,ds+ \int_{\Omega\setminus\Omega_0} dx\int_0^{|\nabla u|^2}a(s)\,ds\biggr) \\ &\geqslant\frac12\int_{\Omega_0} dx\int_0^{|\nabla u|^2}a(s)\,ds, \end{aligned}
\end{equation}
\tag{3.6}
$$
where $\Omega_0=\{x\in\Omega\colon |\nabla u(x)|\geqslant 2t_0\}$, since $a(s)$ is nonnegative on $[0,+\infty)$. By (3.5), for $u\in E$ we have
$$
\begin{equation*}
\begin{aligned} \, \Delta &=\int_{\Omega_0} dx\int_0^{|\nabla u|^2}a(s)\,ds =\int_{\Omega_0} dx\int_0^{4t_0^2}a(s)\,ds+ \int_{\Omega_0} dx\int_{4t_0^2}^{|\nabla u|^2}a(s)\,ds \\ &\geqslant\int_{\Omega_0} dx\int_{4t_0^2}^{|\nabla u|^2}a(s)\,ds\geqslant \int_{\Omega_0} dx\int_{4t_0^2}^{|\nabla u|^2}\frac{C_1}{2^{p-1}}s^{{p}/{2}-1}\,ds \\ &=\frac{2C_1}{p2^{p-1}}\int_{\Omega_0} (|\nabla u|^p-(2t_0)^p)\,dx \\ &\geqslant\frac{2C_1}{p2^{p-1}}\int_\Omega |\nabla u|^p\,dx-\frac{2C_1}{p2^{p-1}} \int_{\Omega\setminus\Omega_0}|\nabla u|^p\,dx- \frac{2C_1(2t_0)^p}{p2^{p-1}}\operatorname{mes}_N\Omega. \end{aligned}
\end{equation*}
\notag
$$
We set $d_1={C_1}/({p2^{p-1}})$. Then
$$
\begin{equation*}
\Delta\geqslant 2d_1\|u\|_{p,1}^p-2d_1(2t_0)^p \operatorname{mes}_N\Omega-2d_1(2t_0)^p \operatorname{mes}_N\Omega=2d_1\|u\|_{p,1}^p-2d_2,
\end{equation*}
\notag
$$
where $d_2=2d_1(2t_0)^p \operatorname{mes}_N\Omega$. From this and (3.6) we obtain for $u\in E$.
To obtain an upper bound for $f_2$ on $E$ we use the assumption (3) of Theorem 1. For any $u\in E$ we have
$$
\begin{equation}
\begin{aligned} \, \notag f_2(u) &=\int_\Omega dx\int_0^{u(x)}g(x,\tau)\,d\tau\leqslant\int_\Omega (M|u(x)|^p+r_1(x)|u|^{p-\theta}+r(x))\,dx \\ \notag &\leqslant M\|u\|_p^p+\|r_1\|_{p/\theta}\|u\|_p^{p-\theta}+\|r\|_1 \\ & \leqslant M\|P\|^p\|u\|_{p,1}^p+\|r_1\|_{p/\theta}\|P\|^{p-\theta}\|u\|_{p,1}^{p-\theta}+\|r\|_1, \end{aligned}
\end{equation}
\tag{3.8}
$$
where $P$ is the operator embedding $E$ in $L_p(\Omega)$. From this and (3.7) we obtain a lower bound on $E$ for the quasipotential $f(u)=f_1(u)-f_2(u)$ of the operator $T=T_1-T_2$.
Thus, for $u\in E$ we have
$$
\begin{equation}
\begin{aligned} \, \notag f(u) &\geqslant d_1\|u\|_{p,1}^p-d_2-M\|P\|^p\|u\|_{p,1}^p -\|r_1\|_{p/\theta}\|P\|^{p-\theta}\|u\|_{p,1}^{p-\theta}-\|r\|_1 \\ &=(d_1-M\|P\|^p)\|u\|_{p,1}^p-d_3\|u\|_{p,1}^{p-\theta}-d_4, \end{aligned}
\end{equation}
\tag{3.9}
$$
where $d_3=\|r_1\|_{p/\theta}\|P\|^{p-\theta}$ and $d_4=d_2+\|r\|_1$.
Since $d_1={C_1}/(p2^{p-1})$, it follows from condition (4) in Theorem 1 that ${d_1>M\|P\|^p}$. This and (3.9) imply that $\lim_{\|u\|\to +\infty}f(u)=+\infty$, that is, the functional $f(u)$ is weakly coercive. It was shown above that the operators $T_1$ and $T_2$ are quasipotential, $T_1$ is radially continuous and monotone and $T_2$ is compact. Hence it follows from Theorem 4 that $f$ is weakly lower semicontinuous on $E$. Since the functional $f$ is weakly coercive and the space $E$ is reflexive, there exists $u_0\in E$ such that $f(u_0)=\inf\{f(u)\colon u\in E\}$ (see [15], Ch. III, § 9, Theorem 9.5). By Theorem 3, if we prove that all the discontinuity points of $T$ are regular, then $Tu_0=0$ and $u_0$ is a point of radial continuity of $T$. Since the operator $T_1$ is radially continuous, it follows that $u_0$ is a point of radial continuity of $T_2$. To complete the proof of Theorem 1 it is necessary to prove that the set
$$
\begin{equation*}
\Omega_1=\{x\in\Omega\colon u_0(x) \text{ is a discontinuity point of the function } g(x,\,\cdot\,)\}
\end{equation*}
\notag
$$
has measure zero in $\mathbb R^N$. As shown below, this is equivalent to the radial continuity of the operator $T_2$ at the point $u_0$.
To prove the regularity of the discontinuity points of $T$ it suffices to establish that for all $u,h\in E$ there exists a nonpositive limit (see Proposition 1). The radial continuity of $T_1$ at $u$ implies that $\lim_{t\to 0+}\langle T_1(u+th)-T_1u, h\rangle=0$ for all $h\in E$. Further,
$$
\begin{equation*}
\langle T_2(u+th)-T_2u, h\rangle=\int_\Omega (g(x,u(x)+th(x))-g(x,u(x)))h(x)\,dx
\end{equation*}
\notag
$$
and the integrand has a limit as $t\to 0+$, which is equal to and is nonnegative almost everywhere on $\Omega$ (by condition (g2)). Here
$$
\begin{equation*}
u(x)_{h(x)}= \begin{cases} u(x)+& \text{ if } h(x)>0,\\ u(x)-& \text{ if } h(x)<0,\\ u(x)& \text{ if } h(x)=0. \end{cases}
\end{equation*}
\notag
$$
By Lebesgue’s theorem we can pass to the limit under the integral sign if we use the bound (3.2). Thus, we have proved that
$$
\begin{equation*}
\lim_{t\to 0+}\langle T(u+th)-Tu, h\rangle\leqslant 0 \quad\forall\, u,h\in E,
\end{equation*}
\notag
$$
and therefore the discontinuity points of $T$ are regular. It remains to prove that $\operatorname{mes}_N\Omega_1=0$.
Assume the contrary, that is, let $\operatorname{mes}_N\Omega_1\neq 0$. By condition (g2) the set $\Omega_1$ differs from only by a subset of measure zero. By condition (g3), for almost all $x\in\Omega$ and $t\in [0,1]$ and each $h\in E$ we have
$$
\begin{equation}
|g(x,u_0(x)+th(x))|\leqslant |d(x)|+k(|u_0(x)|+|h(x)|)^{s-1}\equiv l(x).
\end{equation}
\tag{3.10}
$$
Note that $l(x)$ is integrable on $\Omega$. Since $\operatorname{mes}_N\Omega_1\neq 0$, there exist positive numbers $\varepsilon$ and $\delta$ such that the set
$$
\begin{equation*}
\Omega(\varepsilon)=\{x\in\Omega\colon g(x,u_0(x)+)-g(x,u_0(x)-)>\varepsilon\}
\end{equation*}
\notag
$$
in $\mathbb R^N$ has measure $\delta$. Since $l(x)$ is integrable on $\Omega$, there exists $\eta>0$ for which $\displaystyle\int_A l(x)\,dx<{\varepsilon\delta}/{8}$ if $A$ is a measurable subset of $\Omega$ of measure $\operatorname{mes}_N A<\eta$. Further, there exist a closed set $F\subset\Omega(\varepsilon)$ and an open set $B\supset F$ with closure $\overline B\subset\Omega$ such that $\operatorname{mes}_N F>\operatorname{mes}_N \Omega(\varepsilon)/2=\delta/2$ and $\operatorname{mes}_N (B\setminus F)<\eta$ (see [18]). Let $h\in C_\infty(\overline\Omega)$ be equal to one on $F$ and to zero outside $B$, and let $h(x)\in [0,1]$ if $x\in B\setminus F$. Such a function exists (see [19], Ch. 14, § 2, the lemma). Note that $h\in E$ and almost everywhere on $\Omega$ the limits $\lim_{t\to 0+}g(x,u_0(x)+th(x))h(x)=g(x,u_0(x)_{h(x)})h(x)$ and $\lim_{t\to 0-}g(x,u_0(x)+th(x))h(x)=g(x,u_0(x)_{-h(x)})h(x)$ exist. Hence, taking the bound (3.10) and the integrability of $l(x)$ into account and applying Lebesgue’s theorem, we obtain
$$
\begin{equation*}
\lim_{t\to 0+}\langle T_2(u_0+th), h\rangle\geqslant\int_F g(x,u_0(x)+)\,dx-\int_{B\setminus F}l(x)\,dx
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\lim_{t\to 0-}\langle T_2(u_0+th), h\rangle\leqslant\int_F g(x,u_0(x)-)\,dx+\int_{B\setminus F}l(x)\,dx.
\end{equation*}
\notag
$$
From this, since $\operatorname{mes}_N F>\delta/2$ and $\operatorname{mes}_N (B\setminus F)<\eta$, it follows that
$$
\begin{equation*}
\begin{aligned} \, &\lim_{t\to 0+}\langle T_2(u_0+th), h\rangle - \lim_{t\to 0-}\langle T_2(u_0+th), h\rangle \\ &\qquad\geqslant\int_F (g(x,u_0(x)+)-g(x,u_0(x)-))\,dx -\frac{\varepsilon\delta}{8}-\frac{\varepsilon\delta}{8}\geqslant \frac{\varepsilon\delta}{2}-\frac{\varepsilon\delta}{4}=\frac{\varepsilon\delta}{4}>0, \end{aligned}
\end{equation*}
\notag
$$
which contradicts the radial continuity of $T_2$ at the point $u_0$.
It is interesting to note that if $u\in E$ and
$$
\begin{equation*}
\operatorname{mes}_N\{x\in\Omega\colon u \text{ is a discontinuity point of the function } g(x,\,\cdot\,)\}=0,
\end{equation*}
\notag
$$
then $u(x)$ is a point of radial continuity of $T_2$. This follows immediately from Lebesgue’s dominated convergence theorem if we use the bound (3.10) with $u_0(x)$ replaced by $u(x)$.
Theorem 1 is proved. Proof of Theorem 2. In the operator form problem (1.5), (1.2) in the space ${E=\mathring{W}^1_p(\Omega)}$ reads where $T_\lambda=T_1-\lambda T_2$, $\lambda$ is a positive parameter, and the operators $T_1$ and $T_2$, acting from $E$ to $E^*$, are the same as in the proof of Theorem 1.
It was shown above that $T_1$ and $T_2$ are quasipotential operators with quasipotentials $f_1$ and $f_2$, respectively, where $T_1$ is monotone and radially continuous, and $T_2$ is compact. Hence it follows from Theorem 4 that the quasipotential $f_\lambda=f_1-\lambda f_2$ of the operator $T_\lambda$ is weakly lower semicontinuous on $E$. By the bounds (3.7) and (3.8), taking the condition $M=0$ in Theorem 2 into account, we conclude that $f_\lambda$ is weakly coercive for any $\lambda>0$. Since the space $E$ is reflexive, for each $\lambda>0$ there exists $u_\lambda\in E$ such that $f_\lambda(u_\lambda)=\inf\{f_\lambda(u)\colon u\in E\}$ (see [15], Ch. III, § 9, Theorem 9.5). As in the proof of Theorem 1, the regularity of the discontinuity points of the operator $T_\lambda$ is established using Proposition 1. Hence it follows from Theorem 3 that $u_\lambda$ is a solution to (3.11) and a point of radial continuity of $T_\lambda$. Since the operator $T_1$ is radially continuous on $E$, $u_\lambda$ is a point of radial continuity of $T_2$. It was shown in the proof of Theorem 1 that in this case
$$
\begin{equation*}
\operatorname{mes}\{x\in\Omega\colon u_\lambda(x) \text{ is a point of discontinuity of the function } g(x,\,\cdot\,)\}=0.
\end{equation*}
\notag
$$
Therefore, $u_\lambda(x)$ is a weak semiregular solution of the problem (1.5), (1.2).
By the assumptions of Theorem 2 we have $g(x,0)=0$ almost everywhere on $\Omega$. Therefore, $u(x)=0$ on $\Omega$ is a weak (but not necessarily semiregular) solution of (1.5), (1.2) for any $\lambda>0$. However, only nontrivial solutions of this problem are of interest. Note that the value of $f_\lambda$ at zero in the space $E$ is equal to zero. By the assumption of Theorem 2 there exists an element $\widehat{u}\in E$ for which $f_2(\widehat{u})>0$ (see (1.6)). Therefore, $\lim_{\lambda\to +\infty}f_\lambda(\widehat{u})=-\infty$. Consequently, there exists $\lambda_0>0$ such that $f_\lambda(\widehat{u})<0$ for all $\lambda>\lambda_0$, and therefore $f_\lambda(u_\lambda)<0$. We conclude that for all $\lambda>\lambda_0$ there exists a nontrivial weak semiregular solution $u_\lambda(x)$ of problem (1.5), (1.2). Theorem 2 is proved. 
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\Bibitem{PavPot25} |
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