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# Portugaliae Mathematica

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**Volume 71, Issue 3/4, 2014, pp. 301–395**

**DOI: 10.4171/PM/1954**

Published online: 2014-11-29

Hamiltonian elliptic systems: a guide to variational frameworks

Denis Bonheure^{[1]}, Ederson Moreira dos Santos

^{[2]}and Hugo Tavares

^{[3]}(1) UniversitÃ© libre de Bruxelles, Belgium

(2) Universidade de SÃ£o Paulo, Sao Carlos, Brazil

(3) IST - Universidade de Lisboa, Portugal

Consider a Hamiltonian elliptic system of type \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=H_{v}(u,v) & \text{ in } \Omega\\ -\Delta v=H_{u}(u,v) & \text{ in } \Omega\\ u,v=0 & \text{ on } \partial \Omega \end{array} \right. \end{equation*} where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^{p+1}/(p+1)+|v|^{q+1}/(q+1)$, having subcritical growth, and $\Omega$ is a bounded domain of $\mathbb R^N$, $N\geq 1$. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.

*Keywords: *Hamiltonian elliptic systems, subcritical elliptic problems, qualitative properties, dual method, reduction by inversion, Lyapunov-Schmidt reduction, symmetry properties, multiplicity results, positive and sign-changing solutions, ground state solutions, strongly indefinite functionals

Bonheure Denis, Moreira dos Santos Ederson, Tavares Hugo: Hamiltonian elliptic systems: a guide to variational frameworks. *Port. Math.* 71 (2014), 301-395. doi: 10.4171/PM/1954