Simplicity and boundary behavior of spike sequences for a superlinear problem in plasma physics
Paolo Cosentino, Francesco Malizia
TL;DR
The paper analyzes spike sequences for a nonlinear Grad-Shafranov-type problem in a smooth bounded domain, recasting the equation in terms of a rescaled profile $v_n$ with exploding parameter $\mu_n$. It proves that interior spike points are always simple by a blow-up analysis and a Pohozaev-type contradiction, while boundary spikes are shown to be impossible via a half-space limit and a boundary Pohozaev identity. The interior spikes concentrate at critical points of a Kirchhoff-Routh–type Hamiltonian with unit weights, with the limiting spike profile given by $w_0$ solving $-\Delta w=[w-1]_+^p$ in $\mathbb{R}^N$ and mass $M_{p,0}$; the full asymptotic expansion is governed by sums of Green's functions. These results refine and complement the existing condensation-vanishing framework (Bartolucci–Jevnikar–Wu, 2025) and provide a converse to Wei's 2001 existence results, clarifying the interior- versus boundary-concentration behavior in this plasma-model setting.
Abstract
We prove that spike sequences related to a nonlinear problem of Grad-Shafranov type are always simple and always converge toward interior points of the domain. This sharpens the blow-up analysis carried out by Bartolucci-Jevnikar-Wu [Calc. Var. 2025] and provides a converse to the existence result for spike sequences obtained by Wei [Proc. Edinb. Math. Soc. 2001].