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# Zeitschrift für Analysis und ihre Anwendungen

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Volume 33, Issue 2, 2014, pp. 199–215
DOI: 10.4171/ZAA/1507

Published online: 2014-04-25

On a Singular Class of Hamiltonian Systems in Dimension Two

Abbes Benaissa and Brahim Khaldi

(1) Djillali Liabes University, Sidi Bel Abbes, Algeria
(2) University of Bechar, Algeria

Let $\Omega$ be a bounded domain in $\mathbb{R}^{2}$. In this paper, we consider the following systems of semilinear elliptic equations \text{(S)}\left\{ \begin{alignedat}{2} -\Delta u&=\tfrac{g(v)}{|x|^{a}}&\quad &\hbox{ in }\Omega \\ -\Delta v&=\tfrac{f(u)}{|x|^{b}}&& \hbox{ in }\Omega\\ u&=v=0&& \hbox{ on }\partial\Omega , \end{alignedat} \right. where $a,b\in[0, 2)$ and the functions $f$ and $g$ are nonlinearities having an exponential growth on $\Omega$. This nonlinearity is motivated by suitable Trudinger-Moser inequality with a singular weight. In fact, we prove the existence of a nontrivial solution to (S). For the proof we use a variational argument (a linking theorem).

Keywords: Variational method, Trudinger-Moser inequality, Hamiltonian systems

Benaissa Abbes, Khaldi Brahim: On a Singular Class of Hamiltonian Systems in Dimension Two. Z. Anal. Anwend. 33 (2014), 199-215. doi: 10.4171/ZAA/1507