Multiple solutions of a quasilinear elliptic system involving the nonlinear boundary conditions on exterior domain

作者：Liu, Shuai; Chen, Caisheng^*

来源：Mathematical Methods in the Applied Sciences, 2013, 36(1): 5-17.

DOI：10.1002/mma.2564

• quasilinear p-Laplacian system
• Nehari manifold
• variational methods
• nonlinear boundary conditions
• lack of compactness

In this paper, we consider the multiplicity results of nontrivial nonnegative solutions of the quasilinear p-Laplacian system with the nonlinear boundary conditions: 0.1 -div(a(x)|?u|p-2?u) + b(x)|u|p-2u = d-1Fu(x,u,v), x ?O, -div(a(x)|?v|p-2?v) + b(x)|v|p-2v = d-1Fv(x,u,v), x ?O, a(x)|?u|p-2 ?u ?? = ?h(x)|u|m-2u, x ?G = ?O, a(x)|?v|p-2 ?v ?? = mu H(x)|v|m-2v, x ?G = ?O, where O is a smooth exterior domain in RN(N =3), ? ?? is the outward normal derivative on the boundary G?=??O, and 1 < p < N,1 < m < p < d = p*= Np N-p. By the Nehari manifold and variational methods, we prove that the problem (0.1) has at least two nontrivial nonnegative solutions when the pair of the parameters (?,mu) belongs to a certain subset of R2.

• 出版日期2013-1-15
• 单位河海大学