Positive or sign-changing solutions for a critical semilinear nonlocal equation

TL;DR

This paper proves the existence of infinitely many positive or sign-changing solutions for a critical nonlocal fractional Laplacian equation with radial coefficient functions, under specific asymptotic conditions.

Contribution

It establishes the existence of multiple non-radial solutions for a critical fractional Laplacian problem with variable coefficients, extending previous results to nonlocal operators.

Findings

Existence of infinitely many solutions under certain conditions.

Solutions can be positive or sign-changing.

Results apply to a broad class of radial functions K(|x|).

Abstract

We consider the following critical semilinear nonlocal equation involving the fractional Laplacian $$ (-\Delta)^su=K(|x|)|u|^{2^*_s-2}u,\ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N>2+2s$, $0<s<1$, and $2^*_s=\frac{2N}{N-2s}$. Under some asymptotic assumptions on $K(x)$ at an extreme point, we show that this problem has infinitely many non-radial positive or sign-changing solutions.