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

Mónica Clapp, Alberto Saldaña, Delia Schiera

Abstract

We study the semilinear indefinite elliptic problem \[ -Δu = Q_Ω|u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_Ω= χ_Ω- χ_{\mathbb{R}^N \setminus Ω}$, $Ω\subset \mathbb{R}^N$ 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 $Ω$ 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.

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

Abstract

We study the semilinear indefinite elliptic problem where , is a bounded smooth subset, , and , with corresponding to the sign nonlinearity. Using a variational approach, we investigate the uniqueness or multiplicity of nonnegative solutions depending on the shape of 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 , proving convergence to the whole space as 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.
Paper Structure (19 sections, 33 theorems, 210 equations, 1 figure)

This paper contains 19 sections, 33 theorems, 210 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The variational setting
  3. Ground states and properties of the solutions
  4. Existence of ground states
  5. Properties of the solutions
  6. Starshapedness and Lipschitz regularity
  7. Asymptotic behavior of nonnegative ground states as pto1
  8. Asymptotic behavior of nonnegative ground states as pto2
  9. The structure of the set of nonnegative solutions in the presence of multiple connected components
  10. Nodal solutions
  11. A mountain-pass solution
  12. A least-energy nodal solution
  13. Mountain pass vs. least energy solutions
  14. Nodal symmetric solutions
  15. Proof of main results
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $p\in[1,2)$ and let $\Omega$ be a smooth open bounded subset of $\mathbb{R}^N$.

Figures (1)

  • Figure 1: A comparison between a nonnegative and a nodal solution of \ref{['sublinear']} with $N=1$, $\Omega=[-1,1],$ and $p=1$. Note that the supports are different.

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 62 more