On Concentration Behavior and Multiplicity of Solutions for a System in ℝ^N

We describe a result on the asymptotic behavior of the solutions of a system with two elliptic equations in $\mathbb{R}^{N}$ involving a small parameter. More precisely, we study the system $$\left\{ \begin{aligned} -\varepsilon^{2} \text{div}(a(x) \nabla u)+u & = Q_{u}(u,v)+\frac{\gamma}{2^*} K_u(u,v)\ \ \text{in } \mathbb{R}^N, \\ -\varepsilon^{2} \Delta v + b(x) v & = Q_{v}(u,v)+\frac{\gamma}{2^*} K_v(u,v)\ \ \text{in } \mathbb{R}^N, \\ u,v \in H^{1}(\mathbb{R}^N), &\ u(x),\ v(x)>0\ \ \text{for each } x \in\mathbb{R}^N, \end{aligned} \right.$$ where $2^*=2N/(N-2)$, $N\geq 3$, $\varepsilon>0$, $a$ and $b$ are positive continuous potentials, and $Q$ and $K$ are homogeneous functions with $K$ having critical growth. We use the penalization method for system introduced by C. O. Alves [Local mountain pass for a class of elliptic system, J. Math. Analysis Appl. 335 (2007) 135–150] in order to find a family of solutions $(u_{\varepsilon}, v_{\varepsilon})$ in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ such that, if $\Pi_{\varepsilon,a}$ and $\Pi_{\varepsilon, b}$ are maximum points of $u_{\varepsilon}$ and $v_{\varepsilon}$ respectively, then $$\lim_{\varepsilon \rightarrow 0^+}a(\Pi_{\varepsilon, a}) = \inf_{x \in \mathbb{R}^{N}} a(x) \ \ \ \text{and}\ \ \lim_{\varepsilon \rightarrow 0^+}b(\Pi_{\varepsilon, b})= \displaystyle\inf_{x \in \mathbb{R}^{N}} b(x).$$ Moreover, we relate the number of solutions with the topology of the set where the potentials $a$ and $b$ attain their minima. We consider the subcritical case $\gamma=0$ and the critical case $\gamma=1$.