Advances in Differential Equations

A nonlinear eigenvalue problem in $\Bbb R$ and multiple solutions of nonlinear Schrödinger equation

P. Felmer and J. J. Torres

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Consider the nonlinear Sturm-Liouville eigenvalue problem \begin{align*} u''-Q(x)u & + \lambda(Mu+f(u))=0,\qquad x\in{\mathbb R }, \\ \lim\limits_{|x|\to\infty} u(x) & =\lim\limits_{|x|\to\infty} u'(x) =0, \end{align*} where the potential $Q$ is positive and coercive, the function $f(s)$ behaves like $s^p$, $p>1$, $M$ is a positive constant and $\lambda$ is a positive parameter. When the domain is a bounded interval, Rabinowitz global bifurcation theory applies to this problem, showing the existence of unbounded branches of nontrivial solutions. Even more, Rabinowitz proved that the branches bend back. This last fact has as a consequence a multiplicity result for solutions of a related nonlinear Schr\"odinger equation. In this paper we prove that this result holds true when the domain is ${\mathbb R }$. The main point of the article is the proof that the branches bend back, the place where the noncompactness of ${\mathbb R }$ poses a difficulty.

Article information

Adv. Differential Equations Volume 7, Number 10 (2002), 1215-1234.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Primary: 34B15: Nonlinear boundary value problems
Secondary: 34B40: Boundary value problems on infinite intervals 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05] 47J15: Abstract bifurcation theory [See also 34C23, 37Gxx, 58E07, 58E09]


Felmer, P.; Torres, J. J. A nonlinear eigenvalue problem in $\Bbb R$ and multiple solutions of nonlinear Schrödinger equation. Adv. Differential Equations 7 (2002), no. 10, 1215--1234.

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