Qualitative properties of blowing-up solutions of nonlinear elliptic equations with critical Sobolev exponent

TL;DR

This paper analyzes the qualitative properties of solutions that blow up for a critical nonlinear elliptic equation with a small perturbation, providing eigenvalue estimates, Morse index characterization, and asymptotic behavior under certain conditions.

Contribution

It introduces new estimates on eigenvalues and eigenfunctions, and characterizes the Morse index and asymptotic behavior of blow-up solutions for the critical elliptic equation.

Findings

Eigenvalues and eigenfunctions estimates for blow-up solutions

Morse index of single-bubble solutions is N+1

Asymptotic behavior and nondegeneracy conditions for solutions

Abstract

In this paper, we are concerned with the critical elliptic equation \begin{equation}\label{kx} \left\lbrace\begin{aligned} &-\Delta u=u^{p}+\epsilon \kappa(x)u^{q}\quad\hspace{2mm} \mbox{in}~~\Omega, \\&u>0\quad \quad\quad\quad\quad\quad\quad\quad\hspace{1mm}\hspace{0.5mm}~\mbox{in}~~\Omega \\&u=0\quad \quad\quad\quad\quad\quad\quad\quad\hspace{1mm}\hspace{0.5mm}~\mbox{on}~\partial\Omega, \end{aligned} \right. \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\geq3$, $p=(N+2)/(N-2)$, $1<q<p$, $\epsilon>0$ is a small parameter. If $\kappa(x)=1$, by applying the various identities of derivatives of Green's function and the rescaled functions, with blow-up analysis, we first provide a number of estimates on the first $(N+2)$-eigenvalues and their corresponding eigenfunctions, and prove the qualitative behavior of the eigenpairs $(\lambda_{i,\epsilon}, v_{i,\epsilon})$ to the eigenvalue problem of the elliptic equation \eqref{kx} for $i=1,\cdots,N+2$. As a consequence, we have that the Morse index of a single-bubble solution is $N+1$ if the Hessian matrix of the Robin function is nondegenerate at a blow-up point. Moreover, if $\kappa(x)\in C^2(\overline{\Omega})$, we show that, for $\epsilon>0$ small, the asymptotic behavior of the solutions and nondegeneracy of the solutions for the problem \eqref{kx} under a nondegeneracy condition on the blow-up point of a "mixture" of both the matrix $\kappa(x)$ and Robin function.