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

Abstract

In this paper, we study the existence of nonconstant solutions u ε and their asymptotic behavior (as ε → 0 + ) for the following class of nonlinear elliptic equations in radial form: - ε 2 ( r α | u ′ | β u ′ ) ′ = r γ f ( u ) in ( 0 , R ) , u ′ ( 0 ) = u ′ ( R ) = 0 , where α , β , γ are given real numbers and 0 < R < ∞ . 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.