Infinitely many solutions for anisotropic elliptic equations with variable exponent

In this article, we study the existence and multiplicity of solutions for a class of anisotropic elliptic equations First we establisch that anisotropic space is separable and by using the Fountain theorem, and dual Fountain theorem we prove, under suitable conditions, that the problem (P) admits two sequences of weak solutions.

On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space

PurposeIn the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a non-polynomial growth which does not satisfy any sign condition which is described by an N-uplet of N-functions satisfying the Δ2-condition, within the fulfilling of anisotropic Sobolev-Orlicz space. In addition, the resulting analysis requires the development of some new aspects of the theory in this field. The source term is merely integrable.Design/methodology/approachAn approximation procedure and some priori estimates are used to solve the problem.FindingsThe authors prove the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain. The resulting analysis requires the development of some new aspects of the theory in this field.Originality/valueTo the best of the authors’ knowledge, this is the first paper that investigates the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain.

Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L 1-data

We prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.

Extremal solution and Liouville theorem for anisotropic elliptic equations

<p style='text-indent:20px;'>We study the quasilinear Dirichlet boundary problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber \begin{cases} -Qu = \lambda e^{u}, \text{in}~~ \Omega, \\ u = 0, \qquad \;~~\text{on}~~~~ \partial\Omega, \end{cases} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^{N} $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula>) is a bounded domain, and the operator <inline-formula><tex-math id="M4">\begin{document}$ Q $\end{document}</tex-math></inline-formula>, known as Finsler-Laplacian or anisotropic Laplacian, is defined by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ Qu: = \sum\limits_{i = 1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{\xi_{i}}(\nabla u)). $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Here, <inline-formula><tex-math id="M5">\begin{document}$ F_{\xi_{i}} = \frac{\partial F}{\partial\xi_{i}}(\xi) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ F: \mathbb{R}^{N}\rightarrow [0, +\infty) $\end{document}</tex-math></inline-formula> is a convex function of <inline-formula><tex-math id="M7">\begin{document}$ C^{2}(\mathbb{R}^{N}\setminus\{0\}) $\end{document}</tex-math></inline-formula>, and satisfies certain assumptions. We derive the existence of extremal solution and obtain that it is regular, if <inline-formula><tex-math id="M8">\begin{document}$ N\leq9 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We also concern the Hénon type anisotropic Liouville equation, </p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ -Qu = (F^{0}(x))^{\alpha}e^{u} ~~\text{in} ~~\mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M9">\begin{document}$ \alpha&gt;-2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ N\geq2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ F^{0} $\end{document}</tex-math></inline-formula> is the support function of <inline-formula><tex-math id="M12">\begin{document}$ K: = \{x\in\mathbb{R}^{N}:F(x)&lt;1\} $\end{document}</tex-math></inline-formula>. We obtain the Liouville theorem for stable solutions and finite Morse index solutions for <inline-formula><tex-math id="M13">\begin{document}$ 2\leq N&lt;10+4\alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ 3\leq N&lt;10+4\alpha^{-} $\end{document}</tex-math></inline-formula> respectively, where <inline-formula><tex-math id="M15">\begin{document}$ \alpha^{-} = \min\{\alpha, 0\} $\end{document}</tex-math></inline-formula>.</p>