Some remarks on classical problems and fine properties of Sobolev spaces (Q1909691)

In the first part the eigenvalue problem \[ - \Delta u+ \alpha \rho(x) u= \lambda u\quad \text{in} \quad \Omega,\quad u|\partial\Omega= 0\tag{1} \] is investigated in dependence on the parameter \(\alpha\in \mathbb{R}\). Here \(\Omega\subset \mathbb{R}^n\) is a bounded domain, \(\rho\in L^p(\Omega)\) for some \(p> n/2\), \(\rho\geq 0\). Let \(\lambda_1(\alpha)\) denote the smallest eigenvalue of (1). The author shows necessary and sufficient conditions on the zero set of \(\rho\) for the limit \(\lim_{\alpha\to \infty} \lambda_1(\alpha)\) to be finite or infinite, respectively. In the first case also the (nonclassical) limit problem is studied. An application is given to the nonlinear problem \[ - \Delta u+ \rho(x)|u|^{p- 1} u= \lambda u\quad \text{in} \quad \Omega,\quad u|\partial\Omega= 0,\tag{2} \] \(1< p< (n+ 2)/(n- 2)\), cf. e.g. \textit{T. Ouyang} [Trans. Am. Math. Soc. 331, No. 2, 503-527 (1992; Zbl 0759.35021)]. In the second part a domain variation problem is considered. Let \(\Omega_k\subset \mathbb{R}^n\) be (uniformly) bounded domains converging to a bounded open set \(\Omega_0\) in a suitable weak sense. Let \(c\geq 0\), \(f\in L^p(\mathbb{R}^n)\), \(2n/(n+ 2)< p< \infty\). The author considers the solutions \(u_k\) of \[ - \Delta u_k+ cu_k= f|\Omega_k\quad \text{in} \quad \Omega_k,\quad u_k|\partial \Omega_k= 0,\tag{3} \] and investigates their behaviour for \(k\to \infty\). If \(\Omega_k\subset \Omega_0\), then \(u_k\to u_0\) in \(L^p\), where \(u_0\) is the corresponding solution of the ``limit problem''. In the general case an extra condition on the Sobolev space \(W^{1, 2}_0(\Omega_0)\) is needed.