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Ground state solutions of Hamiltonian elliptic systems in dimension two
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 4 / August 2020
- Published online by Cambridge University Press:
- 12 February 2019, pp. 1737-1768
- Print publication:
- August 2020
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- Article
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The aim of this paper is to study Hamiltonian elliptic system of the form 0.1
where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2
$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).
$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$
