On a class of linearly coupled systems on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> involving asymptotically linear terms
2019; American Institute of Mathematical Sciences; Volume: 18; Issue: 6 Linguagem: Inglês
10.3934/cpaa.2019138
ISSN1553-5258
AutoresEdcarlos D. Silva, José Carlos de Albuquerque, Uberlândio B. Severo,
Tópico(s)Advanced Mathematical Physics Problems
ResumoIn this work we study the existence of positive solutions for the following class of coupled elliptic systems involving nonlinear Schrödinger equations \begin{document}$\left\{ \begin{array}{l}-\Delta u+V_{1}(x)u = f_{1}(u)+\lambda(x)v, & x\in\mathbb{R}^{N},\\-\Delta v+V_{2}(x)v = f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}^{N}, \end{array} \right.$\end{document} where $ N\geq 3 $ and the nonlinearities $ f_{1} $ and $ f_{2} $ are asymptotically linear at infinity. The potentials $ V_{1}(x) $ and $ V_{2}(x) $ are continuous functions which are bounded from below and above. The function $ \lambda(x) $ is continuous and gives us a linear coupling due the terms $ \lambda(x)u $ and $ \lambda(x)v $. Here we employ some variational arguments jointly with a Pohozaev identity.
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