Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

2005/11/08 by Dinu, Teodora Liliana
#35D05 #35J60 #35J70 #58E05 #68T40 #76A02 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.math/0511193

Abstract

We study the boundary value problem -\rm div((|∇ u|^{p_1(x) -2}+|∇ u|^p_2(x)-2)∇ u)=f(x,u) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in \RR^N. We focus on the cases when f_± (x,u)=±(-λ|u|^m(x)-2u+|u|^q(x)-2u), where m(x):=max\p_1(x),p_2(x)\ < q(x) < (N⋅ m(x))/(N-m(x)) for any x∈Ω. In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a \ZZ_2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.

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