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

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.