Existence of solutions for (p(y),q(y))-Laplacian elliptic problem on an exterior domain

Abstract

In this paper, we study the following elliptic problem involving the ( p ⁢ ( y ) , q ⁢ ( y ) {p(y),q(y)} )-Laplacian operator: { - div ⁡ ( a ⁢ ( y ) ⁢ | ∇ ⁡ v | p ⁢ ( y ) - 2 ⁢ ∇ ⁡ v ) + b ⁢ ( y ) ⁢ | v | p ⁢ ( y ) - 2 ⁢ v - div ⁡ ( | ∇ ⁡ v | q ⁢ ( y ) - 2 ⁢ ∇ ⁡ v ) = g ⁢ ( y , v ) , y ∈ Ω , v = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}{-}\operatorname{div}(a(y)|\nabla v|^{p(% y)-2}\nabla v)+b(y)|v|^{p(y)-2}v-\operatorname{div}(|\nabla v|^{q(y)-2}\nabla v% )&\displaystyle=g(y,v),&&\displaystyle y\in\Omega,\\ \displaystyle v&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. with Dirichlet boundary condition in an exterior domain Ω ( ⊂ ℝ n ) {(\subset\mathbb{R}^{n})} with smooth boundary, where 1 < q ⁢ ( y ) < p ⁢ ( y ) < n 1<q(y)<p(y)<n . We prove the existence of solutions in W 0 1 , p ⁢ ( y ) ⁢ ( Ω ) {W^{1,p(y)}_{0}(\Omega)} for the superlinear case by using the Mountain Pass Theorem.