Infinitely many solutions for the prescribed scalar curvature problem with volcano-like curvature
Tuoxin Li, Juncheng Wei, Haidong Yang
TL;DR
The paper tackles the prescribed scalar curvature problem $-\Delta u = K(x) u^{\frac{n+2}{n-2}}$ on $\mathbb{R}^n$ with a volcano-like potential $K(x)$, proving the existence of infinitely many nonradial positive solutions by a Lyapunov–Schmidt reduction around an $k$-bubble configuration. It balances bubble centers and heights, derives a finite-dimensional reduced system via precise projection estimates, and performs a variational reduction to obtain at least two distinct solutions per period with arbitrarily large energy. A key advancement is the demonstration of non-degeneracy of these multi-bump solutions in the full $D^{1,2}(\mathbb{R}^n)$ space and in the symmetric subspace, providing robustness of the construction and enabling further applications. The approach extends the earlier radial results to almost-radial, nonradial potentials and offers a general framework for non-degeneracy analysis in related elliptic problems.
Abstract
In this paper, we consider the following prescribed scalar curvature problem: \begin{equation*} -Δu = K(x) u^{\frac{n+2}{n-2}}, \quad u>0\quad\hbox{in}\quad \mathbb{R}^n, \quad u \in D^{1,2}(\mathbb{R}^n), \end{equation*} where $K(x)$ is a volcano-like positive function such that $$ K(x)= K(r_0)- c_0 | |x|- r_0|^m + O( | |x|- r_0|^{m+θ}),\quad r_0- δ<|x| <r_0+δ$$ with $K(r_0), c_0, δ>0, θ>2, \min \{\frac{n-2}{2}, 2\} < m< n-2$. We first prove the existence of infinitely many positive solutions. A consequence of our proof yields that the infinitely many solutions constructed in \cite{WY} are non-degenerate in the whole $D^{1, 2}(\mathbb{R}^{n})$ space. To our knowledge, it seems to be the first result of infinitely many solutions of prescribed scalar curvature problem when the potential function $K(x)$ is not radial. Our non-degeneracy results are also more complete and improve the result in \cite{GMPS}.