Symmetry breaking and multiplicity for supercritical elliptic Hamiltonian systems in exterior domains

2023/11/30 by Temgoua, Remi Yvant
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.18205

Abstract

We consider positive solutions of the following elliptic Hamiltonian systems \ \beginaligned -Δu+uamp;=a(x)v^p-1~~~in~~A_R
-Δv+vamp;=b(x)u^q-1~~~in~~A_R~~~~~~~~~~~~~~~~~(0.1)
u, vamp;gt;0~~~~~~~~~~~~~~~in~~A_R
u=vamp;=0~~~~~~~~~~~~~~~on~~∂ A_R, \endaligned . where A_R=\x∈ℝ^N: |x|>R\, R>0, N>3, and a(x) and b(x) are positive continuous functions. Under certain symmetry and monotonicity properties on a(x) and b(x), we prove that (0.1) has a positive solution for (p,q) above the standard critical hyperbola, that is, (1)/(p)+(1)/(q)<1-(2)/(N), enjoying the same symmetry and monotonicity properties as the weights a and b. In the case when a(x)=b(x)=1, our result ensures multiplicity as it provides \lfloor (N)/(2)\rfloor-1 (being \lfloor (N)/(2)\rfloor the floor of (N)/(2)) non-radial positive solutions provided that (p-1)(q-1)gt;(1+(2N)/(Λ_H))²((q)/(p)), where Λ_H is the optimal constant in Hardy inequality for the domain A_R.

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