Singular variation of nonlinear eigenvalues. II
[For part I see the second author, ibid. 68, 154-156 (1992; Zbl 0809.35068).] From the text: Let \(M\) be a bounded domain in \(\mathbb{R}^ 3\) with smooth boundary \(\partial M\). Let \(w\) be a fixed point in \(M\). Removing an open ball \(B (\varepsilon; w)\) of radius \(\varepsilon\) with the center \(w\) from \(M\), we get \(M_ \varepsilon = M \backslash \overline {B (\varepsilon; w)}\). For \(p > 1\) and \(\varepsilon > 0\) let \(\lambda (\varepsilon)\) denote the positive number defined by \[ \lambda (\varepsilon) = \inf_{X_ \varepsilon} \int_{M_ \varepsilon} | \nabla u |^ 2 dx, \] where \(X_ \varepsilon = \{u \in H^ 1_ 0 (M_ \varepsilon) : \| u \|_{L^{p+1} (M_ \varepsilon)} = 1\), \(u \geq 0\}\). We put \[ \lambda = \inf_{X} \int_ M | \nabla u |^ 2dx, \] where \(X = \{u \in H^ 1_ 0 (M) : \| u \|_{L^{p + 1} (M)} = 1\), \(u \geq 0\}\). In this paper we show the following Theorem: Assume that the positive solution of \(- \Delta u = \lambda u^ p\) in \(M\) under the Dirichlet condition on \(\partial M\) is unique. Then, there exists a constant \(p^* (M) > 1\) such that for any \(p \in (1, p^* (M))\) we have \(\lambda (\varepsilon) - \lambda = 4 \pi \varepsilon u(w)^ 2 + O(\varepsilon)\) as \(\varepsilon\) tends to zero.
semilinear elliptic boundary value problem
scientific article; zbMATH DE number 549018