Boundary blow up for semilinear elliptic equations with nonlinear gradient terms. (English) Zbl 0840.35034

Let \(G\) be a bounded domain in \(\mathbb{R}^n\), satisfying an interior and an exterior sphere condition. Consider the problem: \[ \Delta u\pm |\nabla u|^q= f(u)\quad\text{in} \quad G,\quad u(x)\to \infty\quad\text{as} \quad x\to \partial G\tag{P}{\(^\pm\)} \] with \(q> 0\) an arbitrary fixed number and \(f\) is a positive, increasing function. The authors are discussing the existence of solutions of the above problem together with their asymptotic behaviour near the boundary. The existence of solutions is proved by the method of upper and lower solutions and the asymptotic behaviour are obtained by means of a phase plane analysis with combination of a comparison method. From these results as a sample for the case \(f(t)= \exp(t)\) the following result follows:
i) If \(q< 2\), then \((\text{P}^\pm)\) has a solution. All solutions satisfy \(u(x)/\log \delta^{- 1}(x)\to 2\) as \(x\to \partial G\).
ii) If \(q\geq 2\), \((\text{P}^+)\) has a solution. All solutions of \((\text{P}^+)\) satisfy \(u(x)/\log \delta^{- 1}(x)\to q\) as \(x\to \partial G\).
iii) If \(q= 2\), \((\text{P}^-)\) is solvable. The asymptotic behaviour of the solutions is the same as in ii), where \(\delta(x)= \text{dist}(x, \partial G)\).


35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J67 Boundary values of solutions to elliptic equations and elliptic systems