Solutions for Fractional Schrödinger Equation Involving Critical Exponent via Local Pohozaev Identities

Abstract

Abstract We consider the following fractional Schrödinger equation involving critical exponent: { ( - Δ ) s ⁢ u + V ⁢ ( y ) ⁢ u = u 2 s * - 1 in ⁢ ℝ N , u > 0 , y ∈ ℝ N , \left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u+V(y)u=u^{2^{*}_{s}-1}&&% \displaystyle\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,&&\displaystyle y\in\mathbb{R}^{N},\end{aligned}\right. where N ≥ 3 {N\geq 3} and 2 s * = 2 ⁢ N N - 2 ⁢ s {2^{*}_{s}=\frac{2N}{N-2s}} is the critical Sobolev exponent. Under some suitable assumptions of the potential function V ⁢ ( y ) {V(y)} , by using a finite-dimensional reduction method, combined with various local Pohazaev identities, we prove the existence of infinitely many solutions. Due to the nonlocality of the fractional Laplacian operator, we need to study the corresponding harmonic extension problem.