Qualitative properties of blowing-up solutions of nonlinear elliptic equations with critical Sobolev exponent
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2025
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| author | Yang, Minbo Zhao, Shunneng |
| author_facet | Yang, Minbo Zhao, Shunneng |
| contents | 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 $Ω$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\geq3$, $p=(N+2)/(N-2)$, $1<q<p$, $ε>0$ is a small parameter. If $κ(x)=1$, 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 $(λ_{i,ε}, v_{i,ε})$ 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 $κ(x)\in C^2(\overlineΩ)$, we show that, for $ε>0$ 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 $κ(x)$ and Robin function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_14401 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Qualitative properties of blowing-up solutions of nonlinear elliptic equations with critical Sobolev exponent Yang, Minbo Zhao, Shunneng Analysis of PDEs 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 $Ω$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\geq3$, $p=(N+2)/(N-2)$, $1<q<p$, $ε>0$ is a small parameter. If $κ(x)=1$, 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 $(λ_{i,ε}, v_{i,ε})$ 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 $κ(x)\in C^2(\overlineΩ)$, we show that, for $ε>0$ 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 $κ(x)$ and Robin function. |
| title | Qualitative properties of blowing-up solutions of nonlinear elliptic equations with critical Sobolev exponent |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.14401 |