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

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.