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

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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