On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets

TL;DR

This paper establishes the precise number of stable solutions with flat or compact support for a class of non-Lipschitz semilinear elliptic equations, depending on domain shape, dimension, and nonlinear exponents.

Contribution

It provides the first exact multiplicity results for stable solutions of these equations considering domain geometry and nonlinear parameters.

Findings

Exact multiplicity of solutions depending on domain and parameters

Use of Pohozhaev's identity and fibering methods in proofs

Clarification of previous results in the literature

Abstract

We prove the exact multiplicity of flat and compact support stable solutions of an autonomous non-Lipschitz semilinear elliptic equation of eigenvalue type according to the dimension N and the two exponents, $0<\alpha<\beta<1$, of the involved nonlinearites. Suitable assumptions are made on the spatial domain $\Omega$ where the problem is formulated in order to avoid a possible continuum of those solutions and, on the contrary, to ensure the exact number of solutions according to the nature of the domain $\Omega$. Our results also clarify some previous works in the literature. The main techniques of proof are a Pohozhaev's type identity and some fibering type arguments in the variational approach.