Higher topological type semiclassical states for fractional nonlinear elliptic equations

Abstract

In this paper, we are concerned with semiclassical states to the following fractional nonlinear elliptic equation $$ \varepsilon^{2s}(-\Delta)^s u + V(x) u=\mathcal{N}(|u|)u \quad \mbox{in } \mathbb R^N, $$ where $0< s < 1$, $\varepsilon> 0$ is a small parameter, $N> 2s$, $V \in C^1\big(\mathbb R^N, \mathbb R^+\big)$ and $\mathcal{N}\in C\big(\mathbb R, \mathbb R^+\big)$. The nonlinearity has Sobolev subcritical, critical or supercritical growth, i.e.\ $\mathcal{N}(t)=t^{p-2}$, $\mathcal{N}(t)=\mu t^{p-2} +t^{2^*_s-2}$ or $\mathcal{N}(t)=t^{p-2} + \lambda t^{r-2}$ for $t \geq 0$, where $2< p< 2^*_s< r$ and $\mu, \lambda> 0$. The fractional Laplacian $(-\Delta)^s$ is characterized as $\mathcal{F}((-\Delta)^{s}u)(\xi)=|\xi|^{2s} \mathcal{F}(u)(\xi)$ for $\xi \in \mathbb R^N$, where $\mathcal{F}$ denotes the Fourier transform. We construct positive semiclassical states and an infinite sequence of sign-changing semiclassical states with higher energies clustering near the local minimum points of the potential $V$. The solutions are of higher topological type, which are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. They correspond to critical points of the underlying energy functional at energy levels where compactness condition breaks down. The proofs are variational and mainly rely on penalization methods, $s$-harmonic extension theories and blow-up arguments along with local type Pohozaev identities.