On some general multiplying solutions results of a Robin problem

TL;DR

This paper proves the existence of solutions for a variable exponent Robin boundary value problem using Ricceri's variational principle, extending the understanding of nonlinear PDEs with variable growth conditions.

Contribution

It introduces new existence results for a class of Robin problems with variable exponents, employing variational methods in weighted Sobolev spaces.

Findings

Existence of solutions established under certain conditions.

Application of Ricceri's variational principle to variable exponent problems.

Extension of solution theory to weighted Sobolev spaces with variable growth.

Abstract

By applying Ricceri's variational principle, we demonstrate the existence of solutions for the following Robin problem \begin{equation*}\left\{ \begin{array}{cc}-\func{div}\left( \omega _{1}(x)\left\vert \nabla u\right\vert^{p(x)-2}\nabla u\right) =\lambda \omega _{2}(x)f(x,u), & x\in \Omega \\ \omega _{1}(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*} in $W_{\omega _{1},\omega _{2}}^{1,p(.)}\left( \Omega \right) $ under some appropriate conditions.