Dancer–Fuc̆ik spectrum for fractional Schrödinger operators with a steep potential well on [formula omitted

Abstract

In this paper, we study Dancer–Fuc̆ik spectrum of the fractional Schrödinger operators which is defined as the set of (α,β)∈R2 such that (−Δ)su+Vλ(x)u=αu++βu−in RNhas a nontrivial solution u, where the potential Vλ has a steep potential well for sufficiently large parameter λ>0. It is allowed that (−Δ)s+Vλ has essential spectrum with finitely many eigenvalues below the infimum of σess(−Δ)s+Vλ. Many difficulties are caused by general nonlocal operators, we develop new techniques to overcome them to construct the first nontrivial curve of Dancer–Fuc̆ik point spectrum by minimax methods, to show some qualitative properties of the curve, and to prove that the corresponding eigenfunctions are foliated Schwartz symmetric. As applications we obtain the existence of nontrivial solutions for nonlinear Schrödinger equations with nonresonant nonlinearity.