Sign-changing solutions for a class of zero mass nonlocal Schr\"odinger equations

TL;DR

This paper proves the existence of sign-changing solutions for a class of fractional Schr"odinger equations with variable potentials, extending classical results to the nonlocal fractional setting and addressing unique analytical challenges.

Contribution

It introduces a novel approach using minimization and deformation techniques to find sign-changing solutions in fractional Schr"odinger equations, including infinitely many solutions when the nonlinearity is odd.

Findings

Existence of at least one sign-changing solution.

Infinitely many solutions when the nonlinearity is odd.

Extension of classical elliptic results to fractional nonlocal equations.

Abstract

We consider the following class of fractional Schr\"odinger equations $$ (-\Delta)^{\alpha} u + V(x)u = K(x) f(u) \mbox{in} \mathbb{R}^{N} $$ where $\alpha\in (0, 1)$, $N>2\alpha$, $(-\Delta)^{\alpha}$ is the fractional Laplacian, $V$ and $K$ are positive continuous functions which vanish at infinity, and $f$ is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when $f$ is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.