On a Singular Class of Hamiltonian Systems in Dimension Two

  • Abbes Benaissa

    Djillali Liabes University, Sidi Bel Abbes, Algeria
  • Brahim Khaldi

    University of Bechar, Algeria

Abstract

Let Ω\Omega be a bounded domain in R2\mathbb{R}^{2}. In this paper, we consider the following systems of semilinear elliptic equations

(S){Δu=g(v)xa in ΩΔv=f(u)xb in Ωu=v=0 on Ω,\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[0,2)a,b\in[0, 2) and the functions ff and gg are nonlinearities having an exponential growth on Ω\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).

Cite this article

Abbes Benaissa, Brahim Khaldi, On a Singular Class of Hamiltonian Systems in Dimension Two. Z. Anal. Anwend. 33 (2014), no. 2, pp. 199–215

DOI 10.4171/ZAA/1507