Existence and multiplicity of positive solutions for a class of elliptic systems
Elliptic systems; Positive solutions; Variational Methods; Penalization Method; Lusternik-Schnirelman category.
In this work we will show the existence and multiplicity of positive solutions for the following class of elliptic systems
\begin{equation}
\left\{\begin{array}{lll}
-\epsilon^{2}\Delta{u}+W(x)u=Q_{u}(u,v) \text{ em } \mathbb{R}^N \\
-\epsilon^{2}\Delta{v}+V(x)v=Q_{v}(u,v) \text{ em } \mathbb{R}^N \\
u,v\in H^{1}(\mathbb{R}^N), u(x), v(x)>0 \text{ para todo } x\in \mathbb{R}^N \end{array}\right.
\tag{$S_{\epsilon}$}
\end{equation}
where $\epsilon>0$ is a parameter, $N \geq 3,$ $W, ~ V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are positive Hölder continuous functions and $Q:\mathbb{R}^2\rightarrow \mathbb{R}$ is a function of class $C^{2}$ with subcritical growth. The main tools used are Variational Methods, Penalization Method, Maximum Principle and Lusternik-Schnirelman Category.