On existence and concentration of solutions for Hamiltonian systems involving fractional operator with critical exponential growth

Augusto C. R. Costa, Bráulio B. V. Maia and Olímpio H. Miyagaki

Mathematische Nachrichten, 2022, vol. 295, issue 8, 1480-1512

Abstract: This paper is concerned with the existence and concentration of ground state solutions for the following class of fractional Schrödinger system (−Δ)1/2u+(λa(x)+1)u=Hv(u,v)inR,u,v∈H1/2(R),(−Δ)1/2v+(λa(x)+1)v=Hu(u,v)inR,u,v∈H1/2(R),\begin{align*} \hspace*{44pt}(-\Delta )^{1/2}u + (\lambda a(x) + 1)u= H_{v}(u,v) \text{ in } \mathbb {R}, \ \ u,v \in H^{1/2}(\mathbb {R}), \hspace*{-44pt}\\ \hspace*{44pt}(-\Delta )^{1/2}v + (\lambda a(x) + 1)v= H_{u}(u,v) \text{ in } \mathbb {R} , \ \ u,v \in H^{1/2}(\mathbb {R}),\hspace*{-44pt} \end{align*}where H has exponential critical growth, λ is a positive parameter and a(x)$a(x)$ has a potential well with int(a−1(0))${\mathrm{int}}\big (a^{-1}(0 ) \big )$ consisting of k disjoint components Ω1,⋯,Ωk$\Omega _{1}, \dots , \Omega _{k}$. The proof relies on variational methods and combines truncation arguments and the Moser iteration technique.

Date: 2022
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