An eigenvalue problem related to the critical Sobolev exponent: variable coefficient case

Abstract

Let ${u_{{\varepsilon}}}$ be a least energy solution to the nearly critical problem \[ -{\Delta} u = c_0 K(x) u^{{p_{{\varepsilon}}}} \; \mbox{in} \; \Omega, \quad u>0 \; \mbox{in} \; \Omega, \quad u|_{{\partial} \Omega} = 0, \] where $\Omega \subset {\mathbb{R}}^N (N \ge 3)$ is a smooth bounded domain, $c_0 = N(N-2)$, ${p_{{\varepsilon}}} = (N+2)/(N-2) - {\varepsilon}$ where ${\varepsilon}>0$ is a small parameter and $K \in C^2(\overline{\Omega})$ is a positive function. Under some assumptions on $K$, we prove several asymptotic estimates of the eigenvalues ${\lambda}_{i,{\varepsilon}}$ and corresponding eigenfunctions $v_{i,{\varepsilon}}$ to the eigenvalue problem \begin{align*} \begin{cases} -{\Delta} v_{i,{\varepsilon}} = {\lambda}_{i,{\varepsilon}} \left( c_0 {p_{{\varepsilon}}} K(x) {u_{{\varepsilon}}}^{{p_{{\varepsilon}}}-1} \right) v_{i,{\varepsilon}} & \mbox{in} \; \Omega, \\ v_{i,{\varepsilon}} = 0 & \mbox{on} \; {\partial}\Omega, \\ \| v_{i,{\varepsilon}} \|_{L^{\infty}(\Omega)} = 1 & \end{cases} \end{align*} as ${\varepsilon} \to 0$, for $i= 2, \cdots, N+1, N+2$.