Periodic problems with double resonance
type: Item
instance of: scholarly article
title: Periodic problems with double resonance
zbMATH Open document ID: 1255.34043
author: Giuseppina Barletta
author: Nikolaos S. Papageorgiou
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2012-08-01T00:00:00Z
review text: The authors study the second-order periodic problem \[ -u''=f(t,u(t)),\quad u(0)=u(b),\;u'(0)=u'(b),\tag{1} \] where \(f\) is \(C^1\) with respect to the second variable; \(f(t,0)\equiv0\); and, at \(0\) and at infinity, the ratio \(\frac{f(t,x)}{x}\) tends to distinct non-zero eigenvalues of the linear \(b\)-periodic problem for \(u''+\lambda u=0\). They show that, under certain supplementary growth assumptions of technical character, the problem has at least six nontrivial solutions. This is done by applying critical point theory to the functional \(\varphi\) whose Euler equation is the equation in (1). Main ingredients of the proof are minimization, the mountain pass theorem and Morse inequalities. First, two constant sign solutions are found by minimization. The mountain pass theorem of a truncated functional yields two further solutions, respectively, above the positive solution and below the negative solution already found. Comparing the critical groups of these critical points with the critical groups at infinity another critical point appears different from the so far found ones and nontrivial. Finally, applying the Morse inequalities, it is shown that one more nontrivial solution must exist. Some mistakes in the paper easy to correct do not harm the reading.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58E30
zbMATH DE Number: 6062889
zbMATH Keywords: periodic problem
zbMATH Keywords: minimization
zbMATH Keywords: mountain pass
zbMATH Keywords: critical groups
zbMATH Keywords: C-condition
zbMATH Keywords: multiple solutions
MaRDI profile type: Publication
full work available at URL: s00030-011-0130-5
OpenAlex ID: W2092023844
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DOI: 10.1007/S00030-011-0130-5
description: scientific article; zbMATH DE number 6062889