Existence and uniqueness of a weak solution for singular weighted Robin problem involving p(.)-biharmonic operator

Year 2024, Volume: 73 Issue: 4, 941 - 956

İsmail Aydın

https://doi.org/10.31801/cfsuasmas.1468665

Abstract

The aim of this paper is to find the existence of solutions for the following class of singular fourth order equation involving the weighted $p(.)$-biharmonic operator:
\begin{equation*}
\left\{
\begin{array}{cc}
\Delta \left( a(x)\left\vert \Delta u\right\vert ^{p(x)-2}\Delta u\right)
=\lambda b(x)\left\vert u\right\vert ^{q(x)-2}u+V(x)\left\vert u\right\vert
^{-\gamma (x)}, x\in \Omega,~ \\

a(x)\left\vert \Delta 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*}
where $%
%TCIMACRO{\U{3a9} }%
%BeginExpansion
\Omega
%EndExpansion
$ is a smooth bounded domain in $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{N}\left( N\geq 2\right) $. Using variational methods, we prove the existence at least one nontrivial weak solution of such a Robin problem in weighted variable exponent second order Sobolev spaces $W_{a}^{2,p(.)}\left(\Omega \right) $ under some appropriate conditions. Finally, we deduce some uniqueness results.

Keywords

Singular problem, p(.)-biharmonic operator, weighted variable exponent Sobolev spaces

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