## Asymptotic estimates and convexity of large solutions to semilinear elliptic equations.(English)Zbl 0889.35028

The author investigates asymptotic estimates and convexity of classical solutions of the boundary value problem $$\Delta u=f(u)$$ in $$D$$, $$u(x)\to \infty$$ as $$x\to \partial D$$. Here $$D\subset \mathbb{R}^N,N>1$$, is a bounded convex smooth domain, $$f(t)$$ is a differentiable positive nondecreasing function on $$[t_0,\infty )$$ satisfying $$f(t_0)=0$$ and $$F(t)^{-1/2}$$ is integrable at infinity, where $$F$$ is the primitive function of $$f$$, $$F(t_0)=0$$. Let $$\delta (x)$$ denote the distance from $$x$$ to the boundary of $$D$$ and $$\Phi (s)$$ be the function defined as $\int _{\Phi (s)}^\infty [2F(t)]^{-1/2} dt=s.$ The author investigates the behavior of $$u(x)-\Phi (\delta (x))$$ near the boundary of $$D$$.
Reviewer: D.Medková (Praha)

### MSC:

 35J60 Nonlinear elliptic equations 35J67 Boundary values of solutions to elliptic equations and elliptic systems

### Keywords:

singular boundary value