Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

Abstract

We consider the supercritical problem -Delta u = vertical bar u vertical bar(p-2) u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, N >= 3, and p >= 2* := 2N/N-2. Bahri and Coron showed that if Omega has nontrivial homology this problem has a positive solution for p = 2*. However, this is not enough to guarantee existence in the supercritical case. For p >= 2(N - 1)/N-3 Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as p increases. More precisely, we show that for p > 2(N - k)/N-k-2 with 1 <= k <= N - 3 there are bounded smooth domains in R-N whose cup-length is k + 1 in which this problem does not have a nontrivial solution. For N = 4, 8, 16 we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents

Symmetries, Hopf fibrations and supercritical elliptic problems

“…Passaseo's result was extended to more general domains by Faya and the authors of this paper in [8]. They also showed that existence may fail even in domains with richer topology.…”

“…Conditions which guarantee that problem (℘ * b ) has a prescribed number of solutions in domains having finite orbits were recently obtained by Faya and the authors. They proved the following result in [8]. Special cases of it were previusly established in [7,10].…”

“…has m pairs of S C -invariant solutions in Ω := h −1 C (Θ); one of them does not change sign and the rest are sign changing.A more general result can be derived from Theorem 3.4, as stated in[8]. Existence in domains with thin spherical holes.…”

Ground states of critical and supercritical problems of Brezis–Nirenberg type

“…Passaseo [16,17] showed that, if Θ is a ball, problem (3.1) does not have a nontrivial solution for λ = 0 and p ≥ 2 * N,k := 2(N −k) N −k−2 . In [5] it is shown that this is also true for doubly starshaped domains.…”

Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold

“…This approach was introduced by Ruf and Srikanth in [26], where the reduction is given by a Hopf map. Reductions may also be performed by means of other maps which preserve the laplacian, or by considering rotational symmetries, or by a combination of both, as has been recently done in [1,11,12,19,20,24,25,30] for different problems.…”