Sublinear elliptic equations with a sharp change of sign in the nonlinearity

TL;DR

This paper investigates sublinear elliptic equations with sign-changing nonlinearities, exploring solution uniqueness, multiplicity, support properties, and connections to overdetermined problems, revealing how solutions behave as the nonlinearity approaches linearity.

Contribution

It introduces a variational framework for analyzing solutions to indefinite elliptic problems with sign-changing nonlinearities, detailing their support, multiplicity, and asymptotic behavior as p approaches 2.

Findings

Solutions have compact support with starshaped and Lipschitz boundaries.

As p approaches 2, solutions converge to the whole space.

The work links these elliptic problems to a two-phase Serrin-type torsion problem.

Abstract

We study the semilinear indefinite elliptic problem \[ -\Delta u = Q_\Omega |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_\Omega = \chi_\Omega - \chi_{\mathbb{R}^N \setminus \Omega}$, $\Omega \subset \mathbb{R}^N$ is a bounded smooth subset, $N \geq 3$, and $1 \leq p < 2$, with $p=1$ corresponding to the sign nonlinearity. Using a variational approach, we investigate the uniqueness or multiplicity of nonnegative solutions depending on the shape of $\Omega$ and the existence of different types of nodal solutions. We also show that all solutions have compact support and analyze how the support of the ground state depends on $p$, proving convergence to the whole space as $p\to 2^{-}$ and identifying some qualitative features such as starshapedness and Lipschitz regularity of the support. We also establish a link between these problems and a two-phase Serrin-type torsion overdetermined problem.