Sign Changing Solutions with Clustered Layers Near the Origin for Singularly Perturbed Semilinear Elliptic Problems On a Ball

Abstract

We study sign changing solutions to equations of the form -∊²∆u+u=f(u) in B, ∂v u=0 on ∂B, where B is the unit ball in ℝ^N (N≥2), ∊ is a positive constant and f(u) behaves like |u|^p-1 u (but not necessarily odd) with 1 < p < (N+2)/(N-2) if N≥3, and 1 < p <∞ if N=2. We show that for any given positive integer n, this problem has a sign changing radial solution v∊(|x|) which changes sign at exactly n spheres ⋃^n/j=1 {|x|=ρ∊/j}, where 0 < ρ∊/1 < ⋯ < ρ∊/n < 1 and as ∊→0, ρ∊/j→0 and v∊→0 uniformly on compact subsets of (0,1]. Moreover, given any sequence ∊k→0, there is a subsequence ∊ki, such that u∊(|x|) converges to some U in C¹loc(ℝ^N) along this subsequence, and U=U(|x|) is a radial sign changing solution of -∆U+U=f(U) in ℝ^N, U ∈ H¹(ℝ^N) with exactly n zeros: 0 < ρ₁ < ⋯ < ρn < ∞, and ∊⁻¹ρ∊/j→pj along the subsequence ∊ki. Hence the sharp layers of the sign changing solution v∊ are clustered near the origin. The same result holds if the Neumann boundary condition is replaced by the Dirichlet boundary condition, or if B is replaced by ℝ^N.