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Nonexistence results for semilinear elliptic equations on metric graphs

Yang Liu, Yong Lin, Haohang Zhang

Abstract

In this paper, we study the nonexistence of solutions to semilinear elliptic equations with a positive potential on metric graphs. In particular, the Laplacian under consideration is of a special type, related to both the vertices and edges of metric graphs. We construct a modified distance function, introduce appropriate test functions, and establish the nonexistence of global solutions under suitable volume growth conditions imposed on the potential. More precisely, the nonnegative solutions or sign-changing solutions to the equations are the trivial zero solutions.

Nonexistence results for semilinear elliptic equations on metric graphs

Abstract

In this paper, we study the nonexistence of solutions to semilinear elliptic equations with a positive potential on metric graphs. In particular, the Laplacian under consideration is of a special type, related to both the vertices and edges of metric graphs. We construct a modified distance function, introduce appropriate test functions, and establish the nonexistence of global solutions under suitable volume growth conditions imposed on the potential. More precisely, the nonnegative solutions or sign-changing solutions to the equations are the trivial zero solutions.

Paper Structure

This paper contains 11 sections, 10 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. The metric graph setting
  4. Two volume measures
  5. The Laplacian on metric graphs
  6. Definition of solutions
  7. Statement of the main results
  8. Proofs of the main results for the elliptic equation
  9. Modified distance functions
  10. Nonexistence for nonnegative global solutions
  11. Nonexistence for sign-changing global solutions

Key Result

Theorem 3.1

Let $\mathcal{G}$ be a metric graph satisfying s3:1. Suppose that $u$ is a nonnegative solution of s1:1 with $\sigma>1$, and that the potential $V:\mathcal{G}\rightarrow\mathbb{R}$ is a positive function satisfying for some constant $C>0$ and every $R\geq R_0=\max\{2j,1\}$, where Then $u\equiv0$ on $\mathcal{G}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Three cases of the distance $d(x, x_0)$ for $x$ moving along the edge. The last case causes singularity of derivative.
  • Figure 2: Illustration of step function, coordinate transformations and the modified distance function $\tilde{d}_e(x,x_0)$ for three cases previously shown by Fig \ref{['fig:distanceFunctions']}.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Proposition 4.1
  • proof
  • ...and 15 more