Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials

Abstract

This paper is concerned with the following fractional Schrödinger equations involving critical exponents: \begin{eqnarray*} (-Δ)^αu+V(x)u=k(x)f(u)+λ|u|^{2_α^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N}, \end{eqnarray*} where $(-\Delta {)}^{\alpha }$ is the fractional Laplacian operator with $\alpha \in (0,1)$, $N\ge 2$, $\lambda$ is a positive real parameter and $2_{\alpha }^{\ast }=2N/(N-2\alpha )$ is the critical Sobolev exponent, $V(x)$ and $k(x)$ are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti-Rabinowitz condition on the subcritical nonlinearity.

References in corpus (3)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• Higher nonlocal problems with bounded potential
• On doubly nonlocal fractional elliptic equations

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• Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
• Degenerate fractional Kirchhoff-type system with magnetic fields and upper critical growth
• On degenerate fractional Schrödinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields