Remarks on eigenvalue problems involving the p(x)-Laplacian

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:352

Article

Fan, Xianling

关键词: p(x)-Laplacian;    Eigenvalue;    Variable exponent Sobolev space;    Constrained variational method;   

DOI  :  10.1016/j.jmaa.2008.05.086

来源: Elsevier

【 摘 要 】

This paper deals with the eigenvalue problem involving the p(x)-Laplacian of the form {(-div(vertical bar del u vertical bar p(x)-2 del u) = lambda vertical bar u vertical bar q(x)-2u in Omega.)(u=0 on partial derivative Omega.) where Omega is a bounded domain in R-N, p is an element of C-0((Omega) over bar), inf(x is an element of Omega) p(x) > 1, q is an element of L-infinity(Omega), 1 <= q(x) <= q(x) + epsilon < p*(x) for x is an element of Omega, epsilon is a positive constant, p*(x) is the Sobolev critical exponent. It is shown that for every t > 0, the problem has at least one sequence of solutions {(u(n,t), lambda(n,t))} such that integral(Omega)1/p(x)vertical bar del u(n,t)vertical bar(p(x)) = t and lambda(n,t) -> infinity as n -> infinity. The principal eigenvalues for the problem in several important cases are discussed especially. The similarities and the differences in the eigenvalue problem between the variable exponent case and the constant exponent case are exposed. Some known results on the eigenvalue problem are extended. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

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