Multiple Solutions for Superlinear Fractional Problems via Theorems of Mixed Type

Abstract

Abstract In this paper, we investigate the existence of multiple solutions for the following two fractional problems: { ( - Δ Ω ) s ⁢ u - λ ⁢ u = f ⁢ ( x , u ) in  ⁢ Ω , u = 0 in  ⁢ ∂ ⁡ Ω    and    { ( - Δ ℝ N ) s ⁢ u - λ ⁢ u = f ⁢ ( x , u ) in  ⁢ Ω , u = 0 in  ⁢ ℝ N ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta_{\Omega})^{s}u-\lambda u&% \displaystyle=f(x,u)&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{in }\partial\Omega\end{% aligned}\right.\qquad\text{and}\qquad\left\{\begin{aligned} \displaystyle(-% \Delta_{\mathbb{R}^{N}})^{s}u-\lambda u&\displaystyle=f(x,u)&&\displaystyle% \text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{in }\mathbb{R}^{N}% \setminus\Omega,\end{aligned}\right. where s ∈ ( 0 , 1 ) {s\in(0,1)} , N > 2 ⁢ s {N>2s} , Ω is a smooth bounded domain of ℝ N {\mathbb{R}^{N}} , and f : Ω ¯ × ℝ → ℝ {f:\bar{\Omega}\times\mathbb{R}\to\mathbb{R}} is a superlinear continuous function which does not satisfy the well-known Ambrosetti–Rabinowitz condition. Here ( - Δ Ω ) s {(-\Delta_{\Omega})^{s}} is the spectral Laplacian and ( - Δ ℝ N ) s {(-\Delta_{\mathbb{R}^{N}})^{s}} is the fractional Laplacian in ℝ N {\mathbb{R}^{N}} . By applying variational theorems of mixed type due to Marino and Saccon and the Linking Theorem, we prove the existence of multiple solutions for the above problems.