The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case
The aim of this paper is to study the trace problem for positive solutions of the equation (1) \(-\Delta u+| u|^{q-1} u=0\) in \(B\), and the associated boundary value problem, where \(B\) is the unit ball centered at the origin in \(\mathbb R^N\) and \(q>1\). The main result states, in the subcritical case \((1<q< {N+1\over N-1})\), that the boundary value problem \(-\Delta u+| u |^{q-1}u=0\) in \(B\), \(\mathrm{tr}_{\partial B}(u)=({\mathcal S},\mu)\) possesses a unique solution for every pair \(({\mathcal S},\mu)\), and the mapping \(({\mathcal S},\mu)\mapsto u\) is isotone and continuous, where \({\mathcal S}\) is a compact subset of \(\partial B,\mu\) is a non-negative Radon measure on \(R=\partial B\setminus {\mathcal S}\) and the trace of \(u\) on \(B\) is denoted by \(\mathrm{tr}_{\partial B}(u)\). The proof uses some results of H. Brézis (personal communication) and \textit{A. Gmira} and \textit{L. Véron} [Duke Math. J. 64, No. 2, 271--324 (1991; Zbl 0766.35015)], and a result on convergence of traces. In the proof of the existence it is shown that the problem possesses a maximal solution \(\overline u\) and a minimal solution \(\underline u\) which can be constructed by certain approximating processes starting with solutions whose boundary trace is in \(L_q(\partial B)\), whereas the uniqueness is established in several steps. There are also treated other special cases. On the other hand, the notion of trace is well defined: If \(u\) is a positive solution of (1), let the set of regular points \({\mathcal R}:=\{\omega\in\partial B:\exists U\) neighborhood of \(\omega\) on \(\partial B\) such that \(\limsup_{r\to 1}\int_U u(r, \sigma)\,d\sigma<\infty\}\). Then \({\mathcal R}\) is open and there exists a unique Radon measure \(\mu\) on \({\mathcal R}\) such that (2) \(\lim_{r\to 1}\int_{\mathcal R} u(r, \sigma)\xi(\sigma)\,d\sigma=\int_{\mathcal R}\xi \,d\mu\), \(\forall\xi\in C_0({\mathcal R})\). The set of singular points \(\partial B\setminus{\mathcal R}\) is denoted by \({\mathcal S}\). The couple \(({\mathcal S},\mu)\) where \(\mu\) is the Radon measure in (2), is called the trace of \(u\) on \(\partial B\) and is denoted by \(\mathrm{tr}_{\partial B}(u)\).
scientific article; zbMATH DE number 1275156