Global perturbative elliptic problems with critical growth in the fractional 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 \cite{MR1801341}, 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 $s\in(0,1/2]$ was already established, with different methods, in \cite{MR3617721} (actually, in \cite{MR3617721} 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 \cite{MR3617721}).