Existence and nonexistence of solutions for the fractional nonlinear Schrödinger equation

Abstract

We consider the equation (Formula presented.) where s ∈ (0, 1), (Formula presented.), N > 2s, (Formula presented.), (Formula presented.) is such that (Formula presented.) and ε > 0 is small. We study the existence and nonexistence of solutions concentrating at a local minimum point of V as ε → 0 without using any symmetry assumption on V. First, we prove that certain type of positive solutions exhibiting peaks does not exist. Then, we study the existence of sign-changing solutions under a suitable configuration of positive and negative peaks. To guarantee the existence, we cannot neglect the interaction between peaks. In particular, by using a minimization argument, we found solutions exhibiting peaks at the vertices of a 2ℓ-regular polygon, such that two adjacent peaks have alternate sign.