Existence and multiplicity solutions for a $\textbf{p(x)}$‎ -‎Laplacian-like equation via the genus theory

Document Type : Original Article

Authors

Mohsen Alimohammady ¹

Shahrbanoo Abolfazli ¹

Asieh Rezvani ²

¹ Department of Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Mazandaran‎, ‎Babolsar‎, ‎Iran

² Department of Mathematics‎, ‎Technical and Vocational University (TVU), Tehran‎, ‎Iran

10.22034/kjm.2025.430387.3085

Abstract

We study the existence and multiplicity of nontrivial weak solutions for the following p(x)-Laplacian-like equation:
\begin{align*}
\begin{cases}
-\vartriangle_{p(x)}^{l}u(x)&= h(x,u(x))+a(x)\vert u(x)\vert^{q(x)-2}u(x),~~~ x\in \Omega,\\
& u(x)=0,~~~~~~ x\in~\partial\Omega,
\end{cases}
\end{align*}
where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$.
By using the variational method and Krasnoselskii's genus theory, we would show the existence and multiplicity of the solutions. Moreover, we would show the closed ness for the set of eigenvalues in the case that $p(x)\equiv p$ is constant

Keywords

p(x)-Laplacian-like

variational method

genus theory

Sobolev space

Subjects

• 35 Partial differential equations

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