Multiple solutions for a class of nonlinear Sturm-Liouville problems on the half line. (English) Zbl 0703.34032

The author considers a class of nonlinear Sturm-Liouville problems. He gives an extensive list of references containing fourty references together with the complete proofs of the following: 7 theorems, 17 remarks, 17 lemmas, 5 propositions and two corollaries which are pieced together to construct essentially the author’s doctoral thesis from the University of Wisconsin Madison (USA) where the reviewer worked as a visiting professor for fourteen months. In the first section the author obtains existence and uniqueness for positive and negative solutions for the problems \[ (1)\quad -u^ n=\lambda r(x)u-F(x,u)u,\quad 0<x<+\infty,\quad u(a)\cos \theta -u'(a) \sin \theta =0,\quad u\in L^ 2[a,\infty), \] where r and F are nonnegative continuous functions, \(a\geq 0\) and \(0\leq \theta \leq \pi /2.\)
In section 2 he obtains the existence of solutions with prescribed number of nodes when the nonlinearitiy is odd near zero. In section 3 the author considers analogous results for radical solutions in higher dimensional class, while in section 4 he considers the partial differential equation \[ (2)\quad \Delta \hat u=\lambda \hat r(x)\hat u(x)-\hat F(x,\hat u(x))\hat u(x),\quad x\in {\mathbb{R}}, \] where \(\hat r:\) \({\mathbb{R}}^ N\to (0,\infty)\) and \(\hat F:\) \({\mathbb{R}}^ N\times {\mathbb{R}}\to (0,\infty)\) are radially symmetric. The author seeks the solution of (2) in the form \(\hat u\in L^ 2({\mathbb{R}}^ N)\cap C^ 2({\mathbb{R}}^ N)\).
Reviewer: F.M.Ragab


34B24 Sturm-Liouville theory
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI


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