Global perturbative elliptic problems with critical growth in the fractional setting - Dipierro, Serena; Lippi, Edoardo Proietti; Valdinoci, Enrico (Topological Methods in Nonlinear Analysis, 2025)
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$$(-Δ)^s u=ε h(x)u^q+u^{2^*_s-1},$$%as long as $ε> 0$ 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 $ε$ 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 thatthe 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}).