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(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 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(0,1/2] was already established, with different methods, in MR3617721 (actually, in MR3617721 it was erroneously believed that the method would have carried through all the fractional cases s(0,1), so, in a sense, the results presented here correct and complete the ones in MR3617721).