Existence of solutions for a class of asymptotically linear fractional Schrödinger equations

Abstract

Abstract In this paper, we focus on studying a fractional Schrödinger equation of the form { ( − Δ ) s u + V ( x ) u = f ( x , u ) in Ω , u > 0 in Ω , u = 0 in R n ∖ Ω , $$ \textstyle\begin{cases}(-\Delta )^{s}u+V(x)u = f(x,u) &\text{in }\Omega , \\ u>0 &\text{in }\Omega , \\ u =0 &\text{in }\mathbb{R}^{n}\setminus \Omega , \end{cases}$$ where 0 2 s $n>2s$ , Ω is a smooth bounded domain in R n $\mathbb{R}^{n}$ , ( − Δ ) s $(-\Delta )^{s}$ denotes the fractional Laplacian of order s, f ( x , t ) $f(x, t)$ is a function in C ( Ω ‾ × R ) $C(\overline{\Omega}\times \mathbb{R})$ , and f ( x , t ) / t $f(x, t)/t $ is nondecreasing in t and converges uniformly to an L ∞ $L^{\infty}$ function q ( x ) $q(x)$ as t approaches infinity. The potential energy V satisfies appropriate assumptions. In the first part of our study, we analyze the asymptotic linearity of the nonlinearity and investigate the occurrence of the bifurcation phenomenon. We employ variational techniques and a “mountain pass” approach in our proof, notable for not assuming the Ambrosetti–Rabinowitz condition or any replacement condition on the nonlinearity. Additionally, we extend our methods to handle cases where the function f ( x , t ) $f(x,t)$ exhibits superlinearity in t at infinity, represented by q ( x ) ≡ + ∞ $q(x)\equiv +\infty $ .

Keywords

• Bifurcation problems
• Fractional partial differential equations
• Fractional Schrödinger equations
• Weak solution
• Stability variational methods