Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$
Yibin Zhang
TL;DR
The paper analyzes boundary-concentrating bubbling solutions to the planarly parameterized equation -Δu = λ u|u|^{p-2} e^{|u|^p} in a bounded domain with Dirichlet boundary conditions, for 0<p<2 and small λ. It employs a Lyapunov–Schmidt reduction built around a C^1-stable critical point of the Kirchhoff–Routh path function to construct solutions with an arbitrary number m of bubbles near the boundary, including sign-changing configurations. A detailed inner-outer expansion, a weighted linear theory, and a nonlinear projected problem are combined with a variational reduction to reduce to a finite-dimensional problem, yielding precise energy expansions and blow-up profiles. The results demonstrate both the multiplicity and the sign-changing nature of bubbling patterns, including symmetry-driven infinite families, and connect to known energy levels 4πm via delicate asymptotics, with comprehensive appendices providing the technical integral identities necessary for the expansions.
Abstract
Very differently from those perturbative techniques of Deng-Musso in [21], we use the assumption of a $C^1$-stable critical point to construct positive or sign-changing solutions with arbitrary $m$ isolated bubbles to the boundary value problem $-Δu=λu|u|^{p-2}e^{|u|^p}$ under homogeneous Dirichlet boundary condition in a bounded, smooth planar domain $Ω$, when $0<p<2$ and $λ>0$ is a small but free parameter. We prove that for any $0<p<1$ the delicate energy expansion of these bubbling solutions always converges to $4πm$ from below, but for any $1<p<2$ the energy always converges to $4πm$ from above, where the latter case sharply recurs a result of De Marchis-Malchiodi-Martinazzi-Thizy in [22] involving concentration and compactness properties at any critical energy level $4πm$ of positive bubbling solutions. A sufficient condition on the intersection between the nodal line of these sign-changing solutions and the boundary of the domain is founded. Moreover, for $λ$ small enough, we prove that when $Ω$ is an arbitrary bounded domain, this problem has not only at least two pairs of bubbling solutions which change sign exactly once and whose nodal lines intersect the boundary, but also a bubbling solution which changes sign exactly twice or three times; when $Ω$ has an axial symmetry, this problem has a bubbling solution which alternately changes sign arbitrarily many times along the axis of symmetry through the domain.