Critical equations with a sharp change of sign in the nonlinearity
Mónica Clapp, Jorge Faya, Alberto Saldaña
TL;DR
This work analyzes a sign-changing, critical-nonlinearity elliptic problem $-\Delta u + \lambda \mathds{1}_\Omega u = Q_\Omega(x)|u|^{2^*-2}u$ in $\mathbb{R}^N$, where $Q_\Omega$ switches between $+1$ and $-1$ across a bounded region $\Omega$. Employing a variational framework with the energy $J_\lambda$ and Nehari constraint, together with Struwe-type concentration-compactness, the paper establishes existence of a least-energy solution for $N\ge 4$ and $\lambda\in(-\Lambda_\Omega,0)$, and nonexistence of a least-energy solution at $\lambda=0$ in strictly starshaped domains via a Pohožaev identity. Topological tools, including cup-length and Lusternik–Schnirelmann category, yield multiplicity results for $\lambda=0$ tied to the topology of holes in $\Omega$, while symmetry considerations imply infinitely many symmetric solutions when $\Omega$ admits large orbits or radial symmetry. The results extend the Brezis–Nirenberg paradigm to a sign-changing critical problem, highlighting the central role of domain geometry and symmetry in concentration behavior and solution multiplicity.
Abstract
We establish the existence and nonexistence of entire solutions to a semilinear elliptic problem whose nonlinearity is the critical power multiplied by a function that takes the value 1 in an open bounded region and the value -1 in its complement. The existence or not of solutions depends on the geometry of the bounded region, in a way analogous to what happens with the classical critical Dirichlet problem in a bounded domain. Our methods are variational and include the use of topological tools.