Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations
The authors study properties of positive solutions of \[ \Delta u+ hu-ku^p= f\tag{1} \] in a (possibly) nonsmooth \(N\)-dimensional domain \(\Omega\), \(N\geq 2\), subject to the condition \[ u(x)\to\infty\quad\text{if}\quad \delta(x):= \text{dist}(x,\partial\Omega)\to 0.\tag{2} \] Here \(p>1\) and \(h\), \(k\), \(f\) are continuous in \(\overline\Omega\) with \(k>0\) and \(f\geq 0\). Positive solutions of (1) satisfying (2) are called large solutions. A central point of this paper is the following localization principle: let \(\Omega\) be a (not necessarily bounded) domain having the graph property and suppose \(u\) is a positive solution of (1) satisfying \(u(x)\to\infty\) locally uniformly as \(x\to\Gamma\), where \(\Gamma\subset\partial\Omega\) is relatively open. If \(v\) is a large solution, then \(v(x)/u(x)\to 1\) locally uniformly as \(x\to\Gamma\). Closely related to this is a uniqueness result for large solutions in bounded domains having the graph property. For bounded Lipschitz domains the authors prove the existence of positive constants \(c_1\leq c_2\) such that the (unique) large solution \(u\) of (1) satisfies \(c_1\delta(x)^{-{2\over p-1}}\leq u(x)\leq c_2\delta(x)^{-{2\over p-1}}\) for all \(x\in\Omega\). This is also a consequence of the localization principle and an existence theorem, obtained for large solutions in bounded domains satisfying the exterior cone condition. If the domain is not Lipschitz, the rate of blow-up at the boundary may be lower. This is proved for domains having a re-entrant cusp in the case \(p\geq(N-1)/(N-3)\). Finally, the authors discuss the dependence of large solutions on the function \(k\) and the domain \(\Omega\).
scientific article; zbMATH DE number 1013373
scientific article