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Non-negative solutions of a sublinear elliptic problem

Julián López-Gómez, Paul H. Rabinowitz, Fabio Zanolin

Abstract

In this paper the existence of solutions, $(λ,u)$, of the problem $$-Δu=λu -a(x)|u|^{p-1}u \quad \hbox{in }Ω, \qquad u=0 \quad \hbox{on}\;\;\partialΩ,$$ is explored for $0 < p < 1$. When $p>1$, it is known that there is an unbounded component of such solutions bifurcating from $(σ_1, 0)$, where $σ_1$ is the smallest eigenvalue of $-Δ$ in $Ω$ under Dirichlet boundary conditions on $\partialΩ$. These solutions have $u \in P$, the interior of the positive cone. The continuation argument used when $p>1$ to keep $u \in P$ fails if $0 < p < 1$. Nevertheless when $0 < p < 1$, we are still able to show that there is a component of solutions bifurcating from $(σ_1, \infty)$, unbounded outside of a neighborhood of $(σ_1, \infty)$, and having $u \gneq 0$. This non-negativity for $u$ cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described.

Non-negative solutions of a sublinear elliptic problem

Abstract

In this paper the existence of solutions, , of the problem is explored for . When , it is known that there is an unbounded component of such solutions bifurcating from , where is the smallest eigenvalue of in under Dirichlet boundary conditions on . These solutions have , the interior of the positive cone. The continuation argument used when to keep fails if . Nevertheless when , we are still able to show that there is a component of solutions bifurcating from , unbounded outside of a neighborhood of , and having . This non-negativity for cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described.
Paper Structure (8 sections, 6 theorems, 138 equations, 11 figures)

This paper contains 8 sections, 6 theorems, 138 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The existence of non-negative solutions for \ref{['1.1']}
  3. The simplest one-dimensional prototype model
  4. The degenerate homoclinic solution
  5. The classical positive solutions
  6. Highly degenerate positive solutions
  7. Multibump solutions
  8. Final comments

Key Result

Theorem 2.1

If 1.2 holds, there is a component, $\mathscr{D}^+$, of the set of nonnegative solutions of 1.1 with $(\sigma_1,\infty)\in \bar{\mathscr{D}}^+ \subset {\mathbb R} \times (\bar{P}\setminus\{0\})$. Moreover, $\mathcal{P}_\lambda(\mathscr{D}^+)= (\sigma_1,\infty)$, and there is a neighborhood, $\hat{O

Figures (11)

  • Figure 1: The surface $z=E(u,v)$.
  • Figure 2: Level lines of the energy function. We have highlighted $[E=0]$.
  • Figure 3: Three solutions of \ref{['1.3']}. The simulation is made for the coefficients $\lambda=1,$$a=2$ and $p=1/2.$ For the initial point we have taken $u_H=64/9$ (according to \ref{['3.8']}) and $\varepsilon=1/2.$ All these solutions have their first inflection point at level $u=u_0^+$ (the abscissa of the nontrivial equilibrium point).
  • Figure 4: Five solutions of \ref{['1.3']} having the same extinction time, $T_H$, for $(\lambda,p)$ constant and $a$ varying. The simulation has been made with $\lambda=1$ and $p=1/2$. For the initial point we have taken $(u_H,0)$ (according to \ref{['3.8']}). The solutions have been computed for the following series of values: $a=1$ ($u_H=16/9$), $a=5/4$ ($u_h=16/9$), $a=3/2$ ($u_H=4$), $a=7/4$ ($u_H=49/9$), $a=2$ ($u_H=64/9$).
  • Figure 5: Different atypical homoclinics of system \ref{['3.3']} for a fixed pair $(a,p)$ and different values of $\lambda$. The simulation is made for the coefficients $a=2$ and $p=2/3.$ The values chosen for $\lambda$ vary from $\lambda=1$ the external orbit) to $\lambda=2$ (the smallest orbit).
  • ...and 6 more figures

Theorems & Definitions (12)

  • Theorem 2.1
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.1
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • ...and 2 more