Singular variation of domains and continuity property of eigenfunction for some semi-linear elliptic equations
Let \(M\) be a bounded domain in \(\mathbb{R}^3\) with smooth boundary \(\partial M\). Let \(w\) be a fixed point in \(M\). By \(B(\varepsilon;w)\) we denote the ball of center \(w\) with radius \(\varepsilon\). We remove \(\overline{B(\varepsilon; w)}\) from \(M\) and we put \(M_\varepsilon= M\setminus\overline{B(\varepsilon; w)}\). We write \(B(\varepsilon; w)= B_i\). Fix \(k\geq 0\) and \(p\in(1,5)\). We put \[ \lambda(\varepsilon)= \inf_{x_\varepsilon}\Biggl( \int_{M_\varepsilon}|\nabla u|^2 dx+ k\int_{\partial M_\varepsilon} u^2d\sigma\Biggr),\tag{1} \] where \[ X_\varepsilon= \{u\in H^1(M_\varepsilon):\| u\|_{L^{p+1} (M_\varepsilon)}= 1,\;u=0\text{ on }\partial M,\;u\geq 0\text{ in }M_\varepsilon\}. \] Then, we know that there exists at least one solution \(u_\varepsilon\) which attains (1). One of the main results of this paper is the following Theorem: Fix \(p\in (1,5)\). Then, there exists a constant \(C\) independent of \(\varepsilon\) such that \[ \sup_{u_\varepsilon\in S_\varepsilon} \sup_{x\in M_\varepsilon} | u_\varepsilon(x)|\leq C<+\infty, \] where \(S_\varepsilon\) is the set of positive solutions of the corresponding boundary value problem, which minimize (1). We also treat the asymptotic behaviour of \(\lambda(\varepsilon)\) and positive solutions \(u_\varepsilon\).
scientific article; zbMATH DE number 840322