Multiplicity results for fractional Laplace problems with critical growth

TL;DR

This paper investigates the existence and multiplicity of solutions for fractional Laplace equations with critical growth, analyzing bifurcation phenomena and the influence of parameters in bounded domains.

Contribution

It provides new multiplicity and bifurcation results for fractional Laplace problems involving critical Sobolev exponents, extending previous theories to nonlinear fractional PDEs.

Findings

Multiple solutions established for certain parameter ranges.

Bifurcation points identified for the fractional Laplace problem.

Conditions under which solutions exist and change multiplicity.

Abstract

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $(-\Delta)^s$ and involving a critical Sobolev term. In particular, we consider $$\begin{cases} (-\Delta)^su=\gamma|u|^{2^*-2}u+f(x,u) & \mbox{in } \Omega u=0 & \mbox{in } \mathbb R^n\setminus \Omega, \end{cases}$$ where $\Omega\subset\mathbb R^n$ is an open bounded set with continuous boundary, $n>2s$ with $s\in(0,1)$, $\gamma$ is a positive real parameter, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $f$ is a Carath\'{e}odory function satisfying different subcritical conditions.