## 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.

### MSC:

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