Extremal parameters, multiplicity and nonexistence for a singular elliptic problem

Abstract

We consider the following class of singular elliptic problems \[ \begin{cases} -Δu = -\dfrac{u}{|u|^{β+1}}χ_{\{|u|>0\}} +λ|u|^{p-1}u, & \text{in }Ω,\\[1mm] u=0, & \text{on }\partialΩ, \end{cases} \] where $\Omega \subset {\mathbb{R}}^N$ is a bounded smooth domain, $N\ge 3$, $0<\beta ,p<1$, and $\lambda >0$. The energy functional is not of class $C^1$ in $H_0^1(\Omega )$, and the standard critical point theory does not apply. We prove that a nontrivial nonnegative weak solution exists if and only if $\lambda$ exceeds a critical value $\bar{\lambda }$, which we identify and locate strictly between two Rayleigh-type parameters $0<{\lambda }_{\ast }<\bar{\lambda }<{\lambda }^{\ast }<+\infty$. This gives a sharp existence threshold. We also prove that for every $\lambda$ in a neighborhood on the left of ${\lambda }^{\ast }$, there exist at least two distinct nontrivial nonnegative weak solutions. The first is a minimizer on the Nehari manifold, with an energy changing sign at ${\lambda }^{\ast }$. The second has a strictly positive energy. For $\lambda <{\lambda }^{\ast }$ sufficiently close to ${\lambda }^{\ast }$, both solutions have positive energy. This contrasts with previous multiplicity results for the same problem, in which the two solutions obtained have energies of opposite signs.

arXiv comment40 pages, 4 figures