Remarks on the Shape of Least-energy Solutions to a Semilinear Dirichlet Problem

Abstract

Structure of least-energy solutions to singularly perturbed semilinear Dirichlet problem ε²Δu - u^α + g(u) = 0 in Ω,u = 0 on ∂Ω, Ω ⊂ ⋅R^N a bounded smooth domain, is precisely studied as ε → 0^+, for 0 < α < 1 and a superlinear, subcritical nonlinearity g(u). It is shown that there are many least-energy solutions for the problem and they are spike-layer solutions. Moreover, the measure of each spike-layer is estimated as ε → 0^+ .