Paper
Qualitative properties of blowing-up solutions of nonlinear elliptic equations with critical Sobolev exponent
Authors
Minbo Yang, Shunneng Zhao
Abstract
In this paper, we are concerned with the critical elliptic equation \begin{equation}\label{kx} \left\lbrace\begin{aligned}
&-Δu=u^{p}+εκ(x)u^{q}\quad\hspace{2mm} \mbox{in}~~Ω,
\\&u>0\quad \quad\quad\quad\quad\quad\quad\quad\hspace{1mm}\hspace{0.5mm}~\mbox{in}~~Ω
\\&u=0\quad \quad\quad\quad\quad\quad\quad\quad\hspace{1mm}\hspace{0.5mm}~\mbox{on}~\partialΩ,
\end{aligned} \right. \end{equation} where is a smooth bounded domain in for , , , is a small parameter. If , 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 -eigenvalues and their corresponding eigenfunctions, and prove the qualitative behavior of the eigenpairs to the eigenvalue problem of the elliptic equation \eqref{kx} for . As a consequence, we have that the Morse index of a single-bubble solution is if the Hessian matrix of the Robin function is nondegenerate at a blow-up point. Moreover, if , we show that, for 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 and Robin function.