Global perturbative elliptic problems with critical growth in the fractional setting

TL;DR

This paper proves the existence of multiple solutions for a fractional elliptic problem with critical growth, extending classical results to fractional Laplacians and correcting previous incomplete work.

Contribution

It establishes the existence of at least two solutions for the fractional problem with critical growth, using perturbative and variational methods, completing and correcting earlier fractional case results.

Findings

Existence of at least two solutions for small perturbation parameter

Solutions are strictly positive under certain conditions

Extends classical elliptic results to fractional Laplacian setting

Abstract

Given $s$, $q\in(0,1)$, and a bounded and integrable function $h$ which is strictly positive in an open set, we show that there exist at least two nonnegative solutions $u$ of the critical problem $$(-\Delta)^s u=\varepsilon h(x)u^q+u^{2^*_s-1},$$ as long as $\varepsilon>0$ is sufficiently small. Also, if $h$ is nonnegative, these solutions are strictly positive. The case $s=1$ was established in [APP00], which highlighted, in the classical case, the importance of combining perturbative techniques with variational methods: indeed, one of the two solutions branches off perturbatively in $\varepsilon$ from $u=0$, while the second solution is found by means of the Mountain Pass Theorem. The case $s\in\left(0,\frac12\right]$ was already established, with different methods, in [DMV17] (actually, in [DMV17] it was erroneously believed that the method would have carried through all the fractional cases $s\in(0,1)$, so, in a sense, the results presented here correct and complete the ones in [DMV17]).