A Liouville type result for bounded, entire solutions to a class of variational semilinear elliptic systems. (English) Zbl 1374.35141

Sourdis, Christos

Differ. Integral Equ. 29, No. 11-12, 1021-1028 (2016).

The author is concerned with the study of entire solutions for the elliptic system \(\Delta u=W_u(u)\) in \({\mathbb R}^n\), \(n\geq 2\), where \(W\in C^1({\mathbb R}^m,{\mathbb R})\), \(m\geq 1\) satisfies \[ -(u-Q)\cdot W_u(u)\leq C(W(u))^{(p-1)/p}\quad\text{ for all } u\in {\mathbb R}^m, \] for some \(Q\in {\mathbb R}^m\), \(p\geq 2\) and some constant \(C>0\).
The main result of the paper reads as follows.
Theorem. Let \(u\in C^2({\mathbb R}^n,{\mathbb R}^m)\) be a solution of \(\Delta u=W_u(u)\) in \({\mathbb R}^n\). If one of the following conditions hold
(i) \(n\geq 4\) and \[ \int_{B_r}W(u) dx=o(r^q)\quad \text{ as }r\to \infty, \] where \(q=n-2-2/(p-1)\);
(ii) \(n=4\), \(p=2\) and \[ \int_{{\mathbb R}^n} W(u)dx<\infty, \] then \(u\) is constant.

Reviewer: Marius Ghergu (Dublin)

MSC:

35J20	Variational methods for second-order elliptic equations
35J91	Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J47	Second-order elliptic systems

Keywords:

Liouville type results; elliptic equation; entire solutions