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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

Manassés de Souza 1,

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 $\gamma >0$.
Keywords: critical points, Trudinger-Moser inequality, Elliptic systems, critical exponents..
Mathematics Subject Classification: Primary: 35J50, 35J55; Secondary: 35Q5.
Citation: Manassés de Souza. On a singular Hamiltonian elliptic systems involving critical growth in dimension two. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1859-1874. doi: 10.3934/cpaa.2012.11.1859
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), 393-413.

[2]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.

[3]

H. Berestycki and P. -L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.

[4]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbbR^2$, Comm. Partial Differential Equations, 17 (1992), 407-435.

[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), 1037-1054.

[6]

D. G. de Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 97-116.

[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), 139-153.

[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), 1754-1776.

[9]

Y. Ding and S. Li, Existence of entire solutions for some elliptic systems, Bulletin of the Australian Mathematical Society, 50 (1994), 501-519.

[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., (). 

[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) 286-304

[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), 467-488.

[13]

J. Hulshot, E. Mitidieri and R. Van der Vorst, Strongly indefinite systems with critical Sobolev exponents, Trans. Amer. Math. Soc., 350 (1998), 2349-2365.

[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), 185-226.

[15]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077. 

[16]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Regional Conf. Ser. in Math., 65, AMS, Providence, RI, 1986.

[17]

N. S. Trudinger, On the embedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.

[18]

G. Zhang and S. Liu, Existence result for a class of elliptic systems with indefinite weights in $\mathbbR^2$, Bound. Value Probl., 2008, Art. ID 217636, 10 pp.

show all references

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), 393-413.

[2]

Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603.

[3]

H. Berestycki and P. -L. Lions, Nonlinear scalar field equations, I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.

[4]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbbR^2$, Comm. Partial Differential Equations, 17 (1992), 407-435.

[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), 1037-1054.

[6]

D. G. de Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc., 343 (1994), 97-116.

[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), 139-153.

[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), 1754-1776.

[9]

Y. Ding and S. Li, Existence of entire solutions for some elliptic systems, Bulletin of the Australian Mathematical Society, 50 (1994), 501-519.

[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., (). 

[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) 286-304

[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), 467-488.

[13]

J. Hulshot, E. Mitidieri and R. Van der Vorst, Strongly indefinite systems with critical Sobolev exponents, Trans. Amer. Math. Soc., 350 (1998), 2349-2365.

[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), 185-226.

[15]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077. 

[16]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations," CBMS Regional Conf. Ser. in Math., 65, AMS, Providence, RI, 1986.

[17]

N. S. Trudinger, On the embedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.

[18]

G. Zhang and S. Liu, Existence result for a class of elliptic systems with indefinite weights in $\mathbbR^2$, Bound. Value Probl., 2008, Art. ID 217636, 10 pp.

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