Concentration phenomena of solutions for some singularly perturbed elliptic equations

We consider the following singularly perturbed elliptic problem{ε2Δu−u+f(u)=0,u>0inB1,∂u∂ν=0on∂B1, where Δ=∑i=1N∂2∂xi2 is the Laplace operator, B1B1 is the unit ball centered at the origin in RNRN (N⩾2N⩾2), ν   denotes the unit outer normal to ∂B1∂B1, ε>0ε>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)(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>0ε>0 is sufficiently small.