# zbMATH — the first resource for mathematics

Symmetry for solutions of semilinear elliptic equations in $$\mathbb{R}^N$$ and related conjectures. (English) Zbl 0940.35084
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.
Reviewer: G.Bottaro (Genova)

##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35C99 Representations of solutions to partial differential equations