## Topological Methods in Nonlinear Analysis

### Hamiltonian elliptic systems with nonlinearities of arbitrary growth

#### Abstract

We study the existence of standing wave solutions for the following class of elliptic Hamiltonian-type systems: $\begin{cases} -\hbar^2\Delta u+ V(x)u = g(v) & \mbox{in } \mathbb{R}^N, \\ -\hbar^2\Delta v+ V(x)v = f(u) & \mbox{in } \mathbb{R}^N, \end{cases}$ with $N\geq 2$, where $\hbar$ is a positive parameter and the nonlinearities $f,g$ are superlinear and can have arbitrary growth at infinity. This system is in variational form and the associated energy functional is strongly indefinite. Moreover, in view of unboundedness of the domain $\mathbb{R}^N$ and the arbitrary growth of nonlinearities we have lack of compactness. We use a dual variational approach in combination with a mountain-pass type procedure to prove the existence of positive solution for $\hbar$ sufficiently small.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 47, Number 2 (2016), 593-612.

Dates
First available in Project Euclid: 13 July 2016

https://projecteuclid.org/euclid.tmna/1468413754

Digital Object Identifier
doi:10.12775/TMNA.2016.018

Mathematical Reviews number (MathSciNet)
MR3559922

Zentralblatt MATH identifier
1362.35119

#### Citation

Cardoso, José A.; do Ó, João M.; Medeiros, Everaldo. Hamiltonian elliptic systems with nonlinearities of arbitrary growth. Topol. Methods Nonlinear Anal. 47 (2016), no. 2, 593--612. doi:10.12775/TMNA.2016.018. https://projecteuclid.org/euclid.tmna/1468413754

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