Large solutions of quasilinear elliptic equations: Existence and qualitative properties. (English) Zbl 0887.35056

The authors are interested in “large” solutions of the quasilinear elliptic equation \[ \text{div}(g(|\nabla u|)\nabla u)= f(u)k(|\nabla u|) \] in a regular domain \(D\subset\mathbb{R}^n\), i.e. in solutions which tend to \(\infty\) close to the boundary \(\partial D\). Among the conditions, which have to be imposed on the nonlinearities, are sufficient fast and monotone growth of \(f\) to \(\infty\) as \(u\to\infty\) (a generalized “Keller-condition”), monotonicity of \(k\), and appropriate growth of \(g\) and \(k\). Besides existence of large solutions, the paper is mainly devoted to a precise analysis of their asymptotic behaviour as \(x\to\partial D\). Under some additional conditions, the most important of which may be the mean convexity of the boundary \(\partial D\) and a generalized “Lazer-McKenna condition”, they show that \(\lim_{x\to\partial D}(u(x)- \Phi(\text{dist}(x, \partial D)))= 0\), where \(\Phi\) is a solution of the corresponding one-dimensional initial value problem \((g(|\Phi'|)\Phi')'= f(\Phi)k(|\Phi'|)\) for \(x>0\), \(\lim_{x\searrow 0} \Phi(x)= \infty\).
Furthermore, uniqueness and convexity questions are addressed.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs