Existence of infinitely many solutions for some superlinear elliptic equations
type: Item
instance of: scholarly article
title: Existence of infinitely many solutions for some superlinear elliptic equations
zbMATH Open document ID: 0716.35028
DOI: 10.1016/0022-247X(90)90044-G
author: Ryuji Kajikiya
author: Kazunaga Tanaka
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ N\) with smooth boundary \(\partial \Omega\) and let g(x,t) be a continuous function which is odd in the t argument and has superlinear growth, i.e., g(x,t)/t\(\to \infty\) as \(| t| \to \infty\) uniformly in \(x\in {\bar \Omega}\). Under an additional, slightly weaker condition on g than by \textit{P. H. Rabinowitz} [Minimax methods in critical point theory with applications to differential equations, Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)], the authors prove by minimax methods that the semilinear problem \[ \Delta u+g(x,u)=0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] has an unbounded sequence of weak solutions in \(H^ 1_ 0(\Omega)\). A similar result is obtained for the problem \[ \Delta u+g(x,u)+f(x)=0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] where \(f\in L^{\mu /(\mu -1)}(\Omega)\), \(\mu >1\), and g satisfies an alternative third requirement.
reviewed by: Philip Schaefer
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 4179920
zbMATH Keywords: infinitely many solutions
zbMATH Keywords: minimax methods
zbMATH Keywords: semilinear problem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4179920