Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains

2019/02/06 by Molle, Riccardo, Passaseo, Donato
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.02314

Abstract

We deals with nonlinear elliptic Dirichlet problems of the form \rm div(|D u|^p-2D u )+f(u)=0 in Ω, u∈ H^1,p₀(Ω) where Ω is a bounded domain in ℝⁿ, n≥ 2, p> 1 and f has supercritical growth from the viewpoint of Sobolev embedding. Our aim is to show that there exist bounded contractible non star-shaped domains Ω, arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if n=2, 12p\over 2-p and Ω=\(ρcosθ,ρsinθ) : |θ|2p\over 2-p there exists s>0 such that the problem has only the trivial solution u≡ 0 for all α∈ (0,π) and s∈ (0, s).

Related

• Nonexistence of solutions for Dirichlet problems with supercritical\n growth in tubular domains
• Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth
• Non existence of solutions for a slightly super-critical elliptic problem with non-power nonlinearity
• On problems with weighted elliptic operator and general growth nonlinearities
• Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains