Existence of solutions of anisotropic elliptic equations with variable exponents of nonlinearity in unbounded domains

L. M. Kozhevnikova ; A. Sh. Kamalеtdinov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2016), pp. 29-41.

Résumé

For anisotropic quasilinear second order elliptic equations in divergence form with a non-standard growth conditions \begin{equation} \sum\limits_{i=1}^{n}(a_i(\mathrm{x},u,\nabla u))_{x_i}-a_0(\mathrm{x},u,\nabla u)=0,\quad \mathrm{x} \in \Omega; \tag{1} \end{equation} in domain $\Omega $ of the space $\mathbb{R}^n,\;\Omega\subsetneq \mathbb{R}^n,\; n \geq 2,$ the Dirichlet problem is considered with homogeneous boundary condition \begin{equation} u\Big|_{\partial\Omega}= 0. \tag{2} \end{equation} It is assumed that the functions $a_i(\mathrm{x},s_0,s_1,\ldots,s_n)$ have an polinomial growth on variable $s_i$ with powers $p_i(\mathrm{x})\in(1,\infty),\;i=0,1,\ldots,n$. As example can be used the equation $$ \sum\limits_{i=1}^{n}(|u_{x_i}|^{p_i(\mathrm{x})-2}u_{x_i})_{x_i}-|u|^{p_0(\mathrm{x})-2}u=\sum\limits_{i=1}^{n}(\phi_i(\mathrm{x}))_{x_i}-\phi_0(\mathrm{x}). $$ In the paper by M. B. Benboubker, E. Azroul, A. Barbara (Quasilinear elliptic problems with nonstandartd growths, Electronic Journal of Differential Equations, 2011) the existence of solutions of the Dirichlet problem in a bounded domain was proved for an isotropic elliptic equations with variable nonlinearities. For isotropic equations with constant power of nonlinearity the existence of solutions of the Dirichlet problem in an arbitrary domain was established by F. E. Browder (Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Nati. Acad. Sci. USA, 1977). The proof is based on an abstract theorem for pseudomonotone operators. In this paper we prove the existence of solutions of the problem (1), (2) without the assumption of boundedness of $\Omega$ and the smoothness of its boundary. Note by $L_{p(\cdot)}(\Omega)$ Lebesgue spaces with variable exponent $p(\mathrm{x})$ and the Luxemburg norm $\|\cdot\|_{p(\cdot)}$. Let the $\overrightarrow{\mathbf p} (\mathrm{x})=(p_0(\mathrm{x}),p_1(\mathrm{x}),...,p_n(\mathrm{x})) \in (L^+_{\infty}(\Omega))^{n+1}\cap(C^+ (\overline{\Omega}))^{n+1}$. The Sobolev–Orlicz space with variable exponents $\mathring {W}_{\overrightarrow{\mathbf p}(\cdot)}^{1}(\Omega)$ is defined as the completion of the space $C_0^{\infty}(\Omega)$ in the norm $$ \|v\|_{\mathring {W}_{\overrightarrow{\mathbf p}(\cdot)}^{1}(\Omega)}=\|v\|_{p_0(\cdot)}+\sum\limits_{i=1}^n\|v_{x_i}\|_{p_i(\cdot)}. $$ It is assumed that \begin{equation} p_+(\mathrm{x})\leq p_0(\mathrm{x}) p_*(\mathrm{x}),\quad \mathrm{x}\in \Omega, \tag{3} \end{equation} where $$p_+(\mathrm{x})=\max\{p_1(\mathrm{x}),p_2(\mathrm{x}),...,p_n(\mathrm{x})\}, \quad p_*(\mathrm{x})=\left\{\begin{array}{ll}\frac{n\overline{p}(\mathrm{x})}{n-\overline{p}(\mathrm{x})}, \overline{p}(\mathrm{x})>n,\\ +\infty, \overline{p}(\mathrm{x})\leq n, \end{array}\right.,$$ $$ \quad\overline{p}(\mathrm{x})={n}\left(\sum\limits_{i=1}^n 1/p_i(\mathrm{x})\right)^{-1}.$$ And it is also assumed that $a_i(\mathrm{x},s_0,\mathrm{s}), \;$ $i=0,\ldots,n,$ $\mathrm{x}\in \Omega,\;$ $\mathbf{s}=(s_0, \mathrm{s})=(s_0,s_1,\ldots,s_{n})\in\mathbb{R}^{n+1}$, are the Caratheodory functions, and there exist positive numbers $\widehat{a}, \overline{a}$ and measurable non-negative function $\phi(\mathrm{x})\in L_1(\Omega),$ $\Phi_i(\mathrm{x})\in L_{p'_i(\cdot)}(\Omega),\;p'_i(\mathrm{x})=p_i(\mathrm{x})/(p_i(\mathrm{x})-1),\;i=0,1,\ldots,n,$ such that for almost all $\mathrm{x}\in\Omega$ and any ${\mathbf s}=(s_0,\mathrm{s})\in\mathbb{R}^{n+1}$ the inequalities hold: \begin{equation} |a_i(\mathrm{x},s_0,\mathrm{s})|\leq \widehat{a} (|s_i|^{p_i(\mathrm{x})-1}+|s_0|^{p_0(\mathrm{x})/p'_i(\mathrm{x})})+\Phi_i(\mathrm{x}),\quad i=0,1,\ldots,n; \tag{4} \end{equation} \begin{equation} \sum\limits_{i=1}^n(a_i(\mathrm{x},s_0,\mathrm{s})-a_i(\mathrm{x},{s}_0,\mathrm{t}))(s_i-t_i)>0,\quad \mathrm{s}\neq \mathrm{t}; \tag{5} \end{equation} \begin{equation} \sum\limits_{i=0}^na_i(\mathrm{x},s_0,\mathrm{s})s_i\geq \overline{a}\sum\limits_{i=0}^n|s_i|^{p_i(\mathrm{x})}-\phi(\mathrm{x}). \tag{6} \end{equation} Elliptic operators ${\mathbf A}:\mathring{W}_{\overrightarrow{\mathbf p}(\cdot)}^{1}(\Omega)\rightarrow \left(\mathring{W}_{\overrightarrow{\mathbf p}(\cdot)}^{1}(\Omega)\right)',$ corresponding to the problem (1), (2), defined by the equation: $$ \langle{\mathbf A}(u),v\rangle=\int\limits_{\Omega}\sum\limits_{i=0}^{n}a_i(\mathrm{x},u,\nabla u)v_{x_i}d\mathrm{x},\quad u(\mathrm{x}), v(\mathrm{x}) \in \mathring{W}^{1}_{\overrightarrow{\mathbf p}(\cdot)} ({\Omega}).$$ It is proved that operator ${\mathbf A}$ is pseudomonotone, bounded and coercitive. On the basis of these properties we prove the theorem. Theorem. If the conditions (3)–(6), there is a generalized solution of the problem (1), (2).

Détail

Mots-clés : anisotropic elliptic equation, existence solution
Keywords: variable exponent, Dirichlet problem, pseudomonotone operator.

Bibliographie

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