An eigenvalue problem related to the critical Sobolev exponent: variable coefficient case. (Q637062)

Let \(N\in \{3,4,\dots \}\), \(c_0=N(N-2)\), \(p_\varepsilon =(N+2)/(N-2)-\varepsilon \), \(\Omega \subset \mathbf R^N\) be a smooth bounded domain, \(K:\Omega \to \mathbf R\) be a given positive \(C^2(\overline \Omega )\) function and \(u_\varepsilon \) denote the positive least energy solution of the problem \( -\Delta u= c_0 K(x)u^{p_\varepsilon } \) in \(\Omega \) with homogeneous Dirichlet boundary condition. In the article the eigenvalues \(\lambda _{i,\varepsilon }\) and eigenfunctions \(v_{i,\varepsilon }\), \(i\in \mathbf N\) of the linear eigenvalue problem \[ -\Delta v=\lambda c_0 p_{\varepsilon } K(x) u_{\varepsilon }^{p_{\varepsilon }-1}v \] in \(\Omega \) with homogeneous boundary conditions are studied. Under the assumptions that (i) \(K\in C^2(\overline \Omega )\), \(K\in (0,1]\), \(K\) attains its maximum in a single nondegenerate interior point \(x_0\in \Omega \) (ii) \(| \Delta K(x_0)| >(N-2)/2| \mu _1| \), where \(\mu _1\) is the smallest eigenvalue of \(Hess K(x_0)\) the asymptotic behaviour as \(\varepsilon \to 0+\) of \(\lambda _{i,\varepsilon }\) and \(v_{i,\varepsilon }\) for \(i=2,\dots , N+2\) is studied. In particular it is proved that the properly scaled functions \(v_{i,\varepsilon }\), \(i=2,\dots , N+2\) converge in \(C^1_{loc}(\overline \Omega \setminus {\{x_0\}})\) as \(\varepsilon \to 0+\) to a certain functions. The article develops the results of \textit{M. Grossi} and \textit{F. Pacella} [Math. Z. 250, 225--256, (2005; Zbl 1122.35087)] where the same statements are proved provided \(K\equiv 1\).