Liouville type theorems for stable solutions of certain elliptic systems

Abstract: We establish Liouville type theorems for elliptic systems with various classes of non-linearities on $\mathbb{R}^N$. We show among other things, that a system has no semi-stable solution in any dimension, whenever the infimum of the derivatives of the corresponding non-linearities is positive. We give some immediate applications to various standard systems, such as the Gelfand, and certain Hamiltonian systems. The case where the infimum is zero is more interesting and quite challenging. We show that any $C^2(\mathbb{R}^N)$ positive entire semi-stable solution of the following Lane-Emden system, {eqnarray*} \hbox{$(N_{\lambda,\gamma})$}50pt {{array}{lcl} \hfill -\Delta u&=&\lambda f(x) \ v^p, \hfill -\Delta v&=&\gamma f(x) \ u^q, {array}.{eqnarray*} is necessarily constant, whenever the dimension $N< 8+3\alpha+\frac{8+4\alpha}{q-1}$, provided $p=1$, $q\ge2$ and $f(x)= (1+|x|^2)^{\frac{\alpha}{2}} $. The same also holds for $p=q\ge2$ provided $N < 2+ \frac{2(2+\alpha)}{p-1} (p+\sqrt{p(p-1)})$. We also consider the case of bounded domains $\Omega\subset\mathbb{R}^N$, where we extend results of Brown et al. \cite{bs} and Tertikas \cite{te} about stable solutions of equations to systems. At the end, we prove a Pohozaev type theorem for certain weighted elliptic systems.