Concentration phenomena of solutions for some singularly perturbed elliptic equations

We consider the following singularly perturbed elliptic problem ⎧⎨ ⎩ ε2 u −u+ f (u) = 0, u>0 in B1, ∂u ∂ν =0 on∂B1, where = N i=1 ∂2 ∂x2 i is the Laplace operator, B1 is the unit ball centered at the origin in RN (N 2), ν denotes the unit outer normal to ∂B1, ε >0 is a constant, and f is a superlinear nonlinearity with subcritical exponent. We will prove that for any given positive integer K (K 1) there exists a solution which is axially symmetric and has exactly K local maximum points located on the axis of symmetry, when ε >0 is sufficiently small. © 2006 Elsevier Inc. All rights reserved.