Juncheng Wei

Chinese University of Hong Kong

On some supercritical problems

We consider two types of supercritical problems. The first one is the so-called Coron’s problem: Δu + up = 0,u > 0 in D = Ω\Bδ(P), u = 0 on ∂D. We show that there exists resonant exponents NN+2-2 < p1 < p2 < ... < pj < ... such that for δ small, Coron’s problem has a solution, provided p > NN+-22- and p = pj. The second problem is nonlinear Schrodinger equation Δu - V (x)u + up = 0,u > 0in Rn, lim|x|→+u(x) = 0 We show that if V (x) = o(|1x|2), then for p > NN+-13-, there is a continuum of positive solution. If V (x) decays fast enough or V (x) is symmetric in N directions, there is also a continuum of solutions when NN+2-2 < p NN+1-3.)