Juncheng Wei Chinese University of Hong Kong
On some supercritical problems
We consider two types of supercritical problems. The first one is the socalled Coron’s problem:
Δu + u^{p} = 0,u > 0 in D = Ω\B_{δ}(P), u = 0 on ∂D. We show that there exists resonant exponents
< p_{1} < p_{2} < ... < p_{j} < ... such that for δ small, Coron’s problem has a solution, provided p > and
p ⁄= p_{j}. The second problem is nonlinear Schrodinger equation Δu  V (x)u + u^{p} = 0,u > 0in R^{n},
lim_{x→+∞}u(x) = 0 We show that if V (x) = o(), then for p > , 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
< p ≤.)
