Ground states for fractional Schrödinger equations involving a critical nonlinearity

Abstract

Abstract This paper is aimed to study ground states for a class of fractional Schrödinger equations involving the critical exponents: ( - Δ ) α ⁢ u + u = λ ⁢ f ⁢ ( u ) + | u | 2 α * - 2 ⁢ u in ⁢ ℝ N , $(-\Delta)^{\alpha}u+u=\lambda f(u)+|u|^{2_{\alpha}^{*}-2}u\quad\text{in }% \mathbb{R}^{N},$ where λ is a real parameter, ( - Δ ) α ${(-\Delta)^{\alpha}}$ is the fractional Laplacian operator with 0 < α < 1 ${0<\alpha<1}$ , 2 α * = 2 ⁢ N N - 2 ⁢ α ${2_{\alpha}^{*}=\frac{2N}{N-2\alpha}}$ with 2 ≤ N ${2\leq N}$ , f is a continuous subcritical nonlinearity without the Ambrosetti–Rabinowitz condition. Based on the principle of concentration compactness in the fractional Sobolev space and radially decreasing rearrangements, we obtain a nonnegative radially symmetric minimizer for a constrained minimization problem which has the least energy among all possible solutions for the above equations, i.e., a ground state solution.