2024/09/30 by Dipierro, Serena, Lippi, Edoardo Proietti, Valdinoci, Enrico · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2409.19896
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)uq+u2^*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 [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 ε from u=0, while the second solution is found by means of the Mountain Pass Theorem. The case s∈(0,\frac12] 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∈(0,1), so, in a sense, the results presented here correct and complete the ones in [DMV17]).