Steady state solutions for the Gierer-Meinhardt system in the whole space

Abstract

We are concerned with the study of positive solutions to the Gierer-Meinhardt system $$ \begin{cases} \displaystyle -Δu+λu=\frac{u^p}{v^q}+ρ(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 3,\\[0.1in] \displaystyle -Δv+μv=\frac{u^m}{v^s} &\quad\mbox{ in }\mathbb{R}^N,\\[0.1in] \end{cases} $$ which satisfy $u(x),v(x)\to 0$ as $|x|\to \infty$. In the above system $p,q,m,s>0$, $\lambda ,\mu \ge 0$ and $\rho \in C({\mathbb{R}}^N)$, $\rho \ge 0$. It is a known fact that posed in a smooth and bounded domain of ${\mathbb{R}}^N$, the above system subject to homogeneous Neumann boundary conditions has positive solutions if $p>1$ and $\sigma =\frac{mq}{(p-1)(s+1)}>1$. In the present work we emphasize a different phenomenon: we see that for $\lambda ,\mu >0$ large, positive solutions with exponential decay exist if $0<\sigma \le 1$. Further, for $\lambda =\mu =0$ we derive various existence and nonexistence results and underline the role of the critical exponents $p=\frac{N}{N-2}$ and $p=\frac{N+2}{N-2}$.