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p-Laplacian problems where the nonlinearity crosses an eigenvalue

p-Laplacian problems where the nonlinearity crosses an eigenvalue DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 13, Number 3, August 2005 pp. 743{753 p-LAPLACIAN PROBLEMS WHERE THE NONLINEARITY CROSSES AN EIGENVALUE Kanishka Perera Department of Mathematical Sciences Florida Institute of Technology, Melbourne, FL 32901, USA Andrzej Szulkin Department of Mathematics, Stockholm University 106 91 Stockholm, Sweden (Communicated by Jean Mawhin) Abstract. Using linking arguments and a cohomological index theory we ob- tain nontrivial solutions of p-Laplacian problems with nonlinearities that in- teract with the spectrum. 1. Introduction. Consider the quasilinear elliptic boundary value problem ¡¢ u = f(x; u) in ­; (1.1) u = 0 on @­; n p¡2 where ­ is a bounded domain in R ; n ¸ 1, ¢ u = div (jruj ru) is the p- Laplacian, 1 < p < 1, and f is a Carath¶eodory function on ­ £ R such that ¸ as t ! 0; f(x; t) ! uniformly in x (1.2) p¡2 jtj t ¸ as jtj ! 1 with ¸ ; ¸ 2= ¾(¡¢ ), the Dirichlet spectrum of ¡¢ on ­. In the semilinear case 0 1 p p p = 2, a well-known theorem of Amann and Zehnder [1] states that this problem has a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete and Continuous Dynamical Systems Unpaywall

p-Laplacian problems where the nonlinearity crosses an eigenvalue

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1078-0947
DOI
10.3934/dcds.2005.13.743
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Abstract

DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 13, Number 3, August 2005 pp. 743{753 p-LAPLACIAN PROBLEMS WHERE THE NONLINEARITY CROSSES AN EIGENVALUE Kanishka Perera Department of Mathematical Sciences Florida Institute of Technology, Melbourne, FL 32901, USA Andrzej Szulkin Department of Mathematics, Stockholm University 106 91 Stockholm, Sweden (Communicated by Jean Mawhin) Abstract. Using linking arguments and a cohomological index theory we ob- tain nontrivial solutions of p-Laplacian problems with nonlinearities that in- teract with the spectrum. 1. Introduction. Consider the quasilinear elliptic boundary value problem ¡¢ u = f(x; u) in ­; (1.1) u = 0 on @­; n p¡2 where ­ is a bounded domain in R ; n ¸ 1, ¢ u = div (jruj ru) is the p- Laplacian, 1 < p < 1, and f is a Carath¶eodory function on ­ £ R such that ¸ as t ! 0; f(x; t) ! uniformly in x (1.2) p¡2 jtj t ¸ as jtj ! 1 with ¸ ; ¸ 2= ¾(¡¢ ), the Dirichlet spectrum of ¡¢ on ­. In the semilinear case 0 1 p p p = 2, a well-known theorem of Amann and Zehnder [1] states that this problem has a

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Discrete and Continuous Dynamical Systems – Unpaywall

Published: Jan 1, 2005

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