3 research outputs found
Existence of solutions for -Laplacian equations
We discuss the problem
\begin{equation*}
\left\{
\begin{array}{ll}
-\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right)
=\lambda (a\left( x\right) \left\vert u\right\vert ^{q(x)-2}u+b(x)\left\vert u\right\vert ^{h(x)-2}u)\text{,} & \text{for }x\in \Omega , \\ u=0\text{,} & \text{for }x\in \partial \Omega .
\end{array}
\right.
\end{equation*}
where is a bounded domain with smooth boundary in and is Lipschitz continuous, and are continuous functions on such that and . We show the existence of at least one nontrivial weak solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem