Some new weighted compact embeddings results and existence of weak
solutions for eigenvalue Robin problem
Abstract
In this paper, 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}, &
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
double weighted variable exponent Sobolev space by applying Mountain
Pass Lemma, Ekeland’s variational principle and Fountain Theorem.