# American Institute of Mathematical Sciences

September  2012, 11(5): 1859-1874. doi: 10.3934/cpaa.2012.11.1859

## On a singular Hamiltonian elliptic systems involving critical growth in dimension two

 1 Departamento de Matem, Brazil

Received  March 2011 Revised  December 2011 Published  March 2012

In this paper we study the existence of nontrivial solutions for the strongly indefinite elliptic system \begin{eqnarray*} -\Delta u + b(x) u = \frac{g(v)}{|x|^\alpha}, v > 0 in R^2, \\ -\Delta v + b(x) v = \frac{f(u)}{|x|^\beta}, u > 0 in R^2, \end{eqnarray*} where $\alpha, \beta \in [0,2)$, $b: \mathbb{R}^2\rightarrow \mathbb{R}$ is a continuous positive potential bounded away from zero and which can be large" at the infinity and the functions $f: \mathbb{R}\rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ behaves like $\exp(\gamma s^2)$ when $|s|\rightarrow+\infty$ for some $\gamma >0$.
Citation: Manassés de Souza. On a singular Hamiltonian elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1859-1874. doi: 10.3934/cpaa.2012.11.1859
##### References:
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##### References:
 [1] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17 (1990), 393.   Google Scholar [2] Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications,, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585.   Google Scholar [3] H. Berestycki and P. -L. Lions, Nonlinear scalar field equations, I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.   Google Scholar [4] D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbbR^2$,, Comm. Partial Differential Equations, 17 (1992), 407.   Google Scholar [5] D. G. de Figueiredo, J. M. do Ó and B. Ruf, Critical and subcritical elliptic systems in dimension two,, Indiana Univ. Math. J., 53 (2004), 1037.   Google Scholar [6] D. G. de Figueiredo and P. L. Felmer, On superquadratic elliptic systems,, Trans. Amer. Math. Soc., 343 (1994), 97.   Google Scholar [7] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbbR^2$ with nonlinearities in the critical growth range,, Calc. Var. Partial Differential Equations, 3 (1995), 139.   Google Scholar [8] M. de Souza and J. M. do Ó, On a class of singular Trudinger-Moser type inequalities and its applications,, Mathematische Nachrichten, 284 (2011), 1754.   Google Scholar [9] Y. Ding and S. Li, Existence of entire solutions for some elliptic systems,, Bulletin of the Australian Mathematical Society, 50 (1994), 501.   Google Scholar [10] J. M. do Ó, Liliane A. Maia and Elves A. B. Silva, Standing wave solutions for system of Schrodinger equations in $\mathbbR^2$ involving critical growth,, to appear., ().   Google Scholar [11] J. M. do Ó, E. Medeiros and U. B. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two,, J. Math. Anal. Appl., 345 (2008), 286.   Google Scholar [12] J. Giacomoni and K. Sreenadh, A multiplicity result to a nonhomogeneous elliptic equation in whole space $\mathbbR^2$,, Adv. Math. Sci. Appl., 15 (2005), 467.   Google Scholar [13] J. Hulshot, E. Mitidieri and R. Van der Vorst, Strongly indefinite systems with critical Sobolev exponents,, Trans. Amer. Math. Soc., 350 (1998), 2349.   Google Scholar [14] V. Kondrat'ev and M. Shubin, Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry,, Operator Theory: Advances and Applications, 110 (1999), 185.   Google Scholar [15] J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.   Google Scholar [16] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Regional Conf. Ser. in Math., (1986).   Google Scholar [17] N. S. Trudinger, On the embedding into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.   Google Scholar [18] G. Zhang and S. Liu, Existence result for a class of elliptic systems with indefinite weights in $\mathbbR^2$,, Bound. Value Probl., (2008).   Google Scholar
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