Weak solutions for fractionalp(x,·)-Laplacian Dirichlet problems with weight

Abstract

The main purpose of this paper is to show the existence of weak solutions for a problem involving the fractional p ⁢ ( x , ⋅ ) {p(x,\cdot\,)} -Laplacian operator of the following form: { ( - Δ p ⁢ ( x , ⋅ ) ) s ⁢ u ⁢ ( x ) + w ⁢ ( x ) ⁢ | u | p ¯ ⁢ ( x ) - 2 ⁢ u = λ ⁢ f ⁢ ( x , u ) in ⁢ Ω , u = 0 in ⁢ ℝ N ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta_{p(x,\cdot\,)})^{s}u(x)+w(x)% \lvert u\rvert^{\bar{p}(x)-2}u&\displaystyle=\lambda f(x,u)&&\displaystyle% \phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% N}\setminus\Omega,\end{aligned}\right. The main tool used for this purpose is the Berkovits topological degree.