Symmetry for solutions of semilinear elliptic equations in \(\mathbb{R}^N\) and related conjectures
Theorem 1. Let \(u\in C^2(\mathbb{R}^n)\) \((n>1)\) such that \(\Delta u+ u(1- u^2)= 0\), suppose that \(\{x\in\mathbb{R}^n: u(x)= 0\}\) is bounded with respect to some \(\nu\in\partial S(0,1)\) and both \(\{x\in\mathbb{R}^n: u(x)> 0\}\) and \(\{x\in\mathbb{R}^n: u(x)<0\}\) are unbounded with respect to \(\nu\), then \(u(x)= \pm\tanh\left({\nu\cdot x+\alpha\over\sqrt 2}\right)\) for every \(x\in\mathbb{R}^n\) and some \(\alpha\in\mathbb{R}^n\). Theorem 2. Let \(u\in C^2(\mathbb{R}^n)\) \((n>1)\), \(u\) bounded, such that \(\Delta u+ f\circ u= 0\), where \(f\) is a locally Lipschitz function on \(\mathbb{R}\) such that there exist \(\mu^-< t_1^-< t_0^-< \mu_0< t^+_0< t^+_1<\mu^+\), \(\delta_0^-,\delta^+_0> 0\) for which \(f\geq 0\) in \(]-\infty,\mu^-[\), \(f<0\) in \(]\mu^-,\mu_0[\), \(f>0\) in \(]\mu_0,\mu^+[\), \(f\leq 0\) in \(]\mu^+,+\infty[\), \(f(t)\leq \delta^-_0(t-\mu_0)\) for all \(t\in [t_0^-,\mu_0[\), \(f(t)\geq \delta^+_0(t- \mu_0)\) for all \(t\in [\mu_0,t^+_0[\), \(f\) is nonincreasing on \(]\mu^-,t^-_1[\) and on \(]t^+_1,\mu^+[\); suppose that \(\{x\in \mathbb{R}^n: u(x)= \mu_0\}\) is bounded with respect to some \(\nu\in\partial S(0,1)\) and both \(\{x\in \mathbb{R}^n: u(x)> \mu_0\}\) and \(\{x\in \mathbb{R}^n: u(x)< \mu_0\}\) are unbounded with respect to \(\nu\), then \(u(x)= g(x\cdot\nu)\) for every \(x\in \mathbb{R}^n\), where \(g''+ f\circ g= 0\) and either \(\lim_{t\to\pm\infty} g(t)= \mu^\pm\) and \(g'(t)> 0\) for all \(t\in\mathbb{R}\) or \(\lim_{t\to\pm\infty} g(t)= \mu^\mp\) and \(g'(t)< 0\) for all \(t\in\mathbb{R}\). Similar results are proved also for reaction-convection-diffusion equations.
scientific article; zbMATH DE number 1370125