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Radial Solutions and a Local Bifurcation Result for a Singular Elliptic Problem with Neumann Condition

Claudio Saccon

Abstract

We study the problem $-Δu=λu-u^{-1}$ with a Neumann boundary condition; the peculiarity being the presence of the singular term $-u^{-1}$. We point out that the minus sign in front of the negative power of $u$ is particularly challenging, since no convexity argument can be invoked. Using bifurcation techniques we are able to prove the existence of solution $(u_λ,λ)$ with $u_λ$ approaching the trivial constant solution $u=λ^{-1/2}$ and $λ$ close to an eigenvalue of a suitable linearized problem. To achieve this we also need to prove a generalization of a classical two-branch bifurcation result for potential operators. Next we study the radial case and show that in this case one of the bifurcation branches is global and we find the asymptotical behavior of such a branch. This results allows to derive the existence of multiple solutions $u$ with $λ$ fixed.

Radial Solutions and a Local Bifurcation Result for a Singular Elliptic Problem with Neumann Condition

Abstract

We study the problem with a Neumann boundary condition; the peculiarity being the presence of the singular term . We point out that the minus sign in front of the negative power of is particularly challenging, since no convexity argument can be invoked. Using bifurcation techniques we are able to prove the existence of solution with approaching the trivial constant solution and close to an eigenvalue of a suitable linearized problem. To achieve this we also need to prove a generalization of a classical two-branch bifurcation result for potential operators. Next we study the radial case and show that in this case one of the bifurcation branches is global and we find the asymptotical behavior of such a branch. This results allows to derive the existence of multiple solutions with fixed.
Paper Structure (4 sections, 5 theorems, 125 equations, 3 figures)

This paper contains 4 sections, 5 theorems, 125 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A local bifurcation result for the singular problem
  3. A variant for the two bifurcation branches theorem for potential operators
  4. A global bifurcation result for radial solutions

Key Result

Theorem 2.1

Let $\hat{\mu}>0$ be an eigenvalue of the following Neumann problem: ($\nu$ denotes the normal to $\partial\Omega$). Then there exists $\rho_0>0$ such that for all $\rho\in]0,\rho_0[$ there exist two distinct pairs $(u_{1,\rho},{\lambda}_{1,\rho})$ and $(u_{2,\rho},{\lambda}_{2,\rho})$ such that, for $i=1,2$:

Figures (3)

  • Figure 1: The different cases
  • Figure 2: Graph of $\Phi_{{\lambda},h}(h)$
  • Figure 3: Bifurcation diagram

Theorems & Definitions (16)

  • Theorem 2.1
  • proof
  • Theorem 3.1: Bifurcation
  • proof
  • Remark 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • Lemma 4.5
  • proof
  • ...and 6 more