Multiple solutions for nonhomogeneous p-Laplacian equations with nonlinear boundary conditions on R+N

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:408

Article

Chen, Caisheng¹  Liu, Lihua¹  Chen, Lin^1,2 

[1] Hohai Univ, Coll Sci, Nanjing 210098, Jiangsu, Peoples R China

[2] Yili Normal Univ, Sch Math & Stat, Yining 835000, Peoples R China

关键词: p-Laplacian equation;    Mountain pass theorem;    Nonlinear boundary condition;    Ekeland's variational principle;    Multiple solutions;   

DOI  :  10.1016/j.jmaa.2013.06.026

来源: Elsevier

【 摘 要 】

In this paper, we study the existence of multiple solutions for the nonlinear boundary value problem {-div(vertical bar del(u)vertical bar(p-2)del u) + V(x)vertical bar u vertical bar(p-2)u = h(x), x is an element of R-+(N), vertical bar del u](p-2) partial derivative u/partial derivative nu = lambda h(1) (x) vertical bar u vertical bar(q-2) u + h(2) (x) vertical bar u vertical bar(r-2)u, x is an element of partial derivative R-+(N) , (0.1) where R-+(N) = {(x', x(N)) is an element of RN-1 X R+} is an upper half space in R-N and 1 < p < N, lambda > 0, and 1 < q < p < r < p(*) = p(N-1)/N-p, and nu denotes the unit outward normal to the boundary partial derivative R-+(N) The functions V (x), h(x), h(1) (x), and h(2)(x) satisfy some suitable conditions. Using the mountain pass theorem and Ekeland's variational principle, we prove that there exist lambda(0), m(0) > 0 such that problem (0.1) admits at least two solutions provided that lambda is an element of (0, lambda(0)) and parallel to h parallel to(p)' <= m(0) < c(1 lambda)((p-1/(r-q)), where the constant c(1) > 0 is independent of lambda > 0. On the other hand, if h(2) = 0, we prove that problem (0.1) admits at least one solution for any lambda > 0 and h is an element of L-p' (R-+(N)). (C) 2013 Elsevier Inc. All rights reserved.

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