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Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

Kenichiro Umezu

Abstract

In this paper, we consider the Laplace equation with a class of indefinite superlinear boundary conditions and study the uniqueness of positive solutions that this problem possesses. Superlinear elliptic problems can be expected to have multiple positive solutions under certain situations. To our end, by conducting spectral analysis for the linearized eigenvalue problem at an unstable positive solution, we find sufficient conditions for ensuring that the implicit function theorem is applicable to the unstable positive one. An application of our results to the logistic boundary condition arising from population genetics is given.

Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition

Abstract

In this paper, we consider the Laplace equation with a class of indefinite superlinear boundary conditions and study the uniqueness of positive solutions that this problem possesses. Superlinear elliptic problems can be expected to have multiple positive solutions under certain situations. To our end, by conducting spectral analysis for the linearized eigenvalue problem at an unstable positive solution, we find sufficient conditions for ensuring that the implicit function theorem is applicable to the unstable positive one. An application of our results to the logistic boundary condition arising from population genetics is given.

Paper Structure

This paper contains 16 sections, 24 theorems, 87 equations, 6 figures.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['thm1']}
  3. Existence
  4. Nonexistence
  5. Instability
  6. Bounds and Asymptotics
  7. Proof of Theorem \ref{['thm2']}
  8. Vanishing positive solutions
  9. Our strategy
  10. Analysis of the second eigenvalue
  11. Applications to indefinite logistic boundary conditions
  12. Known results for positive solutions $u\leq1$
  13. Main results for large positive solutions $u\not\leq1$
  14. Proof of Theorem \ref{['thm:A1']}
  15. Proof of Theorem \ref{['thm:A3']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume that $\int_{\partial\Omega}g(x)<0$. Then, pr:w possesses a positive solution $w_\lambda$ for every $\lambda\in (0,\lambda_1(g))$ and no positive solution for any $\lambda\geq \lambda_1(g)$. Furthermore, $w_{\lambda}$ satisfies the following:

Figures (6)

  • Figure 1: Positive solution set $\{ (\lambda,v_{\lambda})\}$ of \ref{['sppr']} as a smooth curve.
  • Figure 2: Situations admitting \ref{['G']}: (i) $\Gamma_{+}=\bigcup_{j=1}^{k}\gamma_k$, (ii) $\Gamma_{+}=\Gamma_1 \cup \left( \bigcup_{j=2}^{k}\gamma_k \right)$, and (iii) $\Gamma_{-,0}=\gamma_1\cup \gamma_2$.
  • Figure 3: Positive solution set $\{ (\lambda,w_\lambda)\}$ of \ref{['pr:w']} with $g=g_{\delta}$ as a smooth curve in the case that $\delta$ is close to $\delta_{0}$.
  • Figure 4: Positive solution set $\{ (\lambda,w_\lambda)\}$ of \ref{['pr:w:f']} with $g=g_{\delta}$ as a smooth curve in the case that $\delta$ is close to $\delta_{0}$.
  • Figure 5: Cases (i) $\int_{\partial\Omega}r<0$, (ii) $\int_{\partial\Omega}r>0$, and (iii) $\int_{\partial\Omega}r=0$.
  • ...and 1 more figures

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 33 more