# 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 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 $\mathbb{R}^2$, 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 $\mathbb{R}^2$ 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 $\mathbb{R}^2$ 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 $\mathbb{R}^2$, 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 (1970/71), 1077-1092. [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 $\mathbb{R}^2$, 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 $\mathbb{R}^2$, 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 $\mathbb{R}^2$ 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 $\mathbb{R}^2$ 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 $\mathbb{R}^2$, 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 (1970/71), 1077-1092. [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 $\mathbb{R}^2$, Bound. Value Probl., 2008, Art. ID 217636, 10 pp.
 [1] Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455 [2] Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378 [3] Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011 [4] Shengbing Deng, Xingliang Tian. On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022071 [5] Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505 [6] Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure and Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006 [7] Sami Aouaoui, Rahma Jlel. Singular weighted sharp Trudinger-Moser inequalities defined on $\mathbb{R}^N$ and applications to elliptic nonlinear equations. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 781-813. doi: 10.3934/dcds.2021137 [8] Yinbin Deng, Shuangjie Peng, Li Wang. Infinitely many radial solutions to elliptic systems involving critical exponents. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 461-475. doi: 10.3934/dcds.2014.34.461 [9] Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452 [10] Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031 [11] Xiaomei Sun, Yimin Zhang. Elliptic equations with cylindrical potential and multiple critical exponents. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1943-1957. doi: 10.3934/cpaa.2013.12.1943 [12] Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085 [13] Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure and Applied Analysis, 2011, 10 (1) : 327-338. doi: 10.3934/cpaa.2011.10.327 [14] Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 4215-4228. doi: 10.3934/era.2021080 [15] Dongsheng Kang. Quasilinear systems involving multiple critical exponents and potentials. Communications on Pure and Applied Analysis, 2013, 12 (2) : 695-710. doi: 10.3934/cpaa.2013.12.695 [16] Dongsheng Kang, Fen Yang. Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4247-4263. doi: 10.3934/dcds.2012.32.4247 [17] Van Hoang Nguyen. The Hardy–Moser–Trudinger inequality via the transplantation of Green functions. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3559-3574. doi: 10.3934/cpaa.2020155 [18] Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121 [19] Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110 [20] Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212

2020 Impact Factor: 1.916