Multiplicity of nodal solutions for elliptic equations with supercritical exponent in contractible domains

Fast and slow decay solutions for supercritical elliptic problems in exterior domains

“…Except for results in domains involving symmetries or exponents close to critical, see for instance [7,8,10,14,16], solvability of (1.1)- (1.2) in the supercritical case has been a widely open matter, particularly since variational machinery no longer applies, at least in its naturally adapted way for subcritical or critical problems.…”

Multi-Peak Solutions for Super-Critical Elliptic Problems in Domains with Small Holes

“…We finally obtain where I e was defined in (12). The definition of f yields that The easily checked facts that…”

“…While it may be expected that this solution survives a small super-critical perturbation of the exponent as in (1), the indirect variational arguments employed in [2,6] do not seem to give in principle a clue on how to obtain this fact. Solvability when q > N þ2 N À2 in domains ''with topology'' is not true, in general, as shown via counterexamples by Passaseo [11,12], answering negatively the question stated by Brezis [4]. In our recent work [7], we have considered problem (1) in Coron's situation of a domain with a small perforation, and proved solvability whenever e is sufficiently small.…”

Two-bubble solutions in the super-critical Bahri-Coron's problem

“…In [16,17] Passaseo constructs examples that show that the answer is in general negative. Among other results, he finds that for N ≥ 4 there is a topologically nontrivial domain, for which no solution of (1.2) exists if q > N +1 N −3 .…”