On upper bounds for positive solutions of semilinear equations
Suppose that \(E\) is a bounded \(C^{2, \lambda}\) domain in \(\mathbb{R}^d\) and \(L\) is a uniformly elliptic operator in \(E\). The set \({\mathcal U}\) of all positive solutions of the equation \(Lu=\psi(u)\) in \(E\) has been studied by various authors. In a earlier paper [Commun. Pure Appl. Math. 51, 897--936 (1998; Zbl 0934.35049)] the author and \textit{S. E. Kuznetsov} defined, for any Borel subset \(\Gamma\) of \(\partial E\), two such solutions \(u_{\Gamma}\leq w_{\Gamma}\). They also introduced a class of solutions \(u_{\nu}\) in one-to-one correspondence with a certain class \({\mathcal N}_0\) of \(\sigma\)-finite measures \(\nu\) on \(\partial E\). Every \(u\in {\mathcal U}\) is associated with a pair \((\Gamma, \nu)\) where \(\Gamma\) is a Borel subset of \(\partial E\) and \(\nu\in {\mathcal N}_0\) and this pair, denoted by \(\text{ tr}(u)\), is called the fine boundary trace of \(u\). Let \(u\oplus v\) denote the maximal solution dominated by \(u+v\). The domain \(E\) is said to belong to \({\mathcal E}_{L, \psi}\) if the condition \(\text{ tr}(u)=(\Gamma, \nu)\) implies \(u\leq w_{\Gamma}\oplus u_{\nu}\). In a recent memoir [``Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation''. Mem. Am. Math. Soc. 798, 121 p. (2004; Zbl 1274.35139)] \textit{B. Mselati} proved that all bounded \(C^4\) domains belong to \({\mathcal E}_{\Delta, \psi}\) where \(\Delta\) is the Laplacian and \(\psi(u)=u^2\). In the present paper, the author, by modifying Mselati's argument, extends the previous result to \(\psi(u)=u^{\alpha}\) with \(1<\alpha\leq 2\) and all bounded \(C^{2, \lambda}\) domains.
scientific article; zbMATH DE number 2091739