Existence and asymptotic behavior of solutions for a class of nonlinear elliptic equations with Neumann condition

In this paper, we study the existence of nonconstant solutions u(epsilon) and their asymptotic behavior (as epsilon -> 0(+)) for the following class of nonlinear elliptic equations in radial form: {-epsilon(2)(r(alpha)vertical bar mu'vertical bar(beta)mu')' = r(gamma) f(u) in (0, R), u' (0) = u' (r) = 0 where alpha, beta, gamma are given real numbers and 0 < R < infinity. We use a version of the Mountain Pass Theorem and the main difficulty is to prove that the solutions so obtained are not constants. For that matter, we have to carryout a careful analysis of the solutions of some Dirichlet problems associated with the Neumann problem. (c) 2004 Elsevier Ltd. All rights reserved.