Differential and Integral Equations

Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth

João Marcos do Ó and José Carlos de Albuquerque

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In this paper, we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schrödinger equations with square root of the Laplacian $$ \begin{cases} (-\Delta)^{ \frac 12 } u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R},\\ (-\Delta)^{ \frac 12 } v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}, \end{cases} $$ where the nonlinearities $f_{1}(s)$ and $f_{2}(s)$ have exponential critical growth of the Trudinger-Moser type, the potentials $V_{1}(x)$ and $V_{2}(x)$ are nonnegative and periodic. Moreover, we assume that there exists $\delta\in (0,1)$ such that $\lambda(x)\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$. We are also concerned with the existence of ground states when the potentials are asymptotically periodic. Our approach is variational and based on minimization technique over the Nehari manifold.

Article information

Differential Integral Equations, Volume 31, Number 5/6 (2018), 403-434.

First available in Project Euclid: 23 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J50: Variational methods for elliptic systems 35B33: Critical exponents 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]


do Ó, João Marcos; de Albuquerque, José Carlos. Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth. Differential Integral Equations 31 (2018), no. 5/6, 403--434. https://projecteuclid.org/euclid.die/1516676436

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