Blow-up in exterior domains: existence and star-shapedness
The authors study the problem \[ \text{div}(g(| \nabla u|)\nabla u)=f(u) k(| \nabla u|),\quad x\in\Omega \subset \mathbb{R}^N,\quad N\geq 2,\quad \lim _{x\to \partial \Omega}u(x)=+\infty. \tag{1} \] Let the functions \(g, f\) and \(k\) satisfy the following conditions: \(f\in C^1((t_0,+\infty),\mathbb{R}^+)\), \(f(t^+_0)=0\), \(f(+\infty)=+\infty\), \(f'\geq 0, f'(t^+_0)<+\infty\); \(g\in C^2([0,+\infty),\mathbb{R}^+)\), \(G(t):=(g(t)t)'>0\), \(t\geq 0\); \(k\in C^1([0,+\infty),\mathbb{R}^+)\). There exist \(\tau >-1\) and \(q\leq \tau +2\) such that for large \(t\) the following relations hold \(C_1t^{\tau}<G(t)<C_2t^{\tau}\), \(C_1 t^q< k(t)<C_2t^q\) for some positive constants \(C_1\) and \(C_2\). Denote by \(F(t):=\int_{t_0}^t f(s)ds, H(t):=\int _0^t G(| s|)s/ k(| s|)ds\) and suppose that \[ \int _0^{+\infty} \frac{1}{H^{-1}(F(t))} dt<+\infty \tag{2} \] Theorem. Let \(\Omega\) be a (the exterior of contractible) bounded \(C^{2,\alpha}\) domain. Under the above conditions there exists a solution \(u\in C^{2,\alpha}(\Omega)\), \(u>t_0\) of problem (1). In the case \(\Omega\) is an exterior domain, every solution to the problem satisfies \(u(x)\to t_0\) as \(| x| \to +\infty\). The authors also show that condition (2) is necessary for the existence of solutions to the problem (1) exploding at the boundary. The authors also investigate the star-shapedness of the level sets of large solutions in exterior domains in the two-dimensional case and in the \(N\)-dimensional case when equation (1) is semilinear.
scientific article; zbMATH DE number 1182357
scientific article