Communications of theKorean Mathematical SocietyCKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2023; 38(4): 1045-1061

Online first article October 16, 2023      Printed October 31, 2023

https://doi.org/10.4134/CKMS.c220308

Multiplicity results of critical local equation related to the genus theory

M. Alimohammady, A. Rezvani, C. Tunc

University of Mazandarn; Qaemshahr Branch, Islamic Azad University; Van Yuzuncu Yil university

Abstract

Using variational methods, Krasnoselskii's genus theory and symmetric mountain pass theorem, we introduce the existence and multiplicity of solutions of a parameteric local equation. At first, we consider the following equation $\begin{cases} -div [a(x, |\nabla u|) \nabla u] = \mu (b(x) |u|^{s(x) -2} - |u|^{r(x) -2})u & \text{in} ~~\Omega,\\ u=0 & \text{on}~~ \partial \Omega, \end{cases}$ where $\Omega \subseteq \mathbb{R}^N$ is a bounded domain, $\mu$ is a positive real parameter, $p$, $r$ and $s$ are continuous real functions on $\bar{\Omega}$ and $a(x, \xi)$ is of type $|\xi|^{p(x) -2}$. Next, we study boundedness and simplicity of eigenfunction for the case $a(x, |\nabla u|) \nabla u= g(x) | \nabla u|^{p(x) -2}\nabla u$, where $g\in L^{\infty}(\Omega)$ and $g(x) \geq 0$ and the case $a(x, |\nabla u|) \nabla u= (1+ \nabla u|^2)^{\frac{p(x) -2}{2}} \nabla u$ such that $p(x) \equiv p$.

Keywords: $p(x)$-Laplacian, modular function, genus theory

MSC numbers: Primary 58B34, 58J42, 81T75

Supported by: This work was financially supported by KRF 2003-041-C20009.