Two-bubble nodal solutions for slightly subcritical Fractional Laplacian

TL;DR

This paper proves the existence of solutions with two bubbles for a slightly subcritical fractional Laplacian problem, extending classical results to a nonlocal setting with small perturbations.

Contribution

It introduces new existence results for two-bubble nodal solutions in a fractional Laplacian context with subcritical perturbations.

Findings

Existence of two-bubble nodal solutions established.

Extension of classical results to fractional Laplacian case.

Addresses nonlocal effects in subcritical elliptic problems.

Abstract

In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian \begin{equation*} \left\{\aligned &(-\Delta)^su=|u|^{p-1-\varepsilon}u\ \ \mbox{in}\ \Omega &u=0\ \mbox{on}\ \partial\Omega, \endaligned \right. \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N>2s$, $0<s<1$, $ p=\frac{N+2s}{N-2s}$ and $\varepsilon>0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) \cite{Bartsch1}.