Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

2023/09/06 by Alberto Boscaggin, Boscaggin, Alberto, Francesca Colasuonno +5
Mathematics · #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2309.03029

Abstract

We deal with the following semilinear equation in exterior domains -Δu + u = a(x)|u|^p-2u, u∈ H¹₀(A_R), where A_R := \x∈ℝ^N: |x|>R\, N≥ 3, R>0. Assuming that the weight a is positive and satisfies some symmetry and monotonicity properties, we exhibit a positive solution having the same features as a, for values of p>2 in a suitable range that includes exponents greater than the standard Sobolev critical one. In the special case of radial weight a, our existence result ensures multiplicity of nonradial solutions. We also provide an existence result for supercritical p in nonradial exterior domains.

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