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 \(D\).


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