Some new weighted compact embeddings results and existence of weak solutions for eigenvalue Robin problem

TL;DR

This paper establishes new weighted compact embedding results and proves the existence and multiplicity of weak solutions for a nonlinear eigenvalue Robin boundary value problem using variational methods.

Contribution

It introduces novel weighted compact embeddings and applies advanced variational principles to demonstrate solution existence and multiplicity for a nonlinear Robin problem.

Findings

Proved weighted compact embedding theorems.

Established existence of weak solutions for the Robin problem.

Demonstrated multiple solutions under certain conditions.

Abstract

By applying Mountain Pass Lemma, Ekeland's and Ricceri's variational principle, Fountain Theorem, we prove the existence and multiplicity of solutions for the following Robin problem \begin{equation*} \left\{ \begin{array}{cc} -\text{div}\left( a(x)\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right) =\lambda b(x)\left\vert u\right\vert ^{q(x)-2}u, & x\in \Omega \\ a(x)\left\vert \nabla u\right\vert ^{p(x)-2}\frac{\partial u}{\partial \upsilon }+\beta (x)\left\vert u\right\vert ^{p(x)-2}u=0, & x\in \partial \Omega , \end{array} \right. \end{equation*} under some appropriate conditions in the space $W_{a,b}^{1,p(.)}\left( \Omega \right).$