# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021156
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## Elliptic systems involving Schrödinger operators with vanishing potentials

 1 Departamento de Matemática y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile 2 Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil 3 Departamento de Matemática y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

* Corresponding author: Pedro Ubilla

Received  February 2021 Revised  July 2021 Early access November 2021

Fund Project: The second author was partially supported by Proyecto código 041933UL POSTDOC, Dirección de Investigación, Científica y Tecnológica, DICYT. The third author was partially supported by FONDECYT Grant 1181125, 1161635, 1171691

We prove the existence of a bounded positive solution of the following elliptic system involving Schrödinger operators
 $\begin{equation*} \left\{ \begin{array}{cll} -\Delta u+V_{1}(x)u = \lambda\rho_{1}(x)(u+1)^{r}(v+1)^{p}&\mbox{ in }&\mathbb{R}^{N}\\ -\Delta v+V_{2}(x)v = \mu\rho_{2}(x)(u+1)^{q}(v+1)^{s}&\mbox{ in }&\mathbb{R}^{N},\\ u(x),v(x)\to 0& \mbox{ as}&|x|\to\infty \end{array} \right. \end{equation*}$
where
 $p,q,r,s\geq0$
,
 $V_{i}$
is a nonnegative vanishing potential, and
 $\rho_{i}$
has the property
 $(\mathrm{H})$
introduced by Brezis and Kamin [4].As in that celebrated work we will prove that for every
 $R> 0$
there is a solution
 $(u_R, v_R)$
defined on the ball of radius
 $R$
centered at the origin. Then, we will show that this sequence of solutions tends to a bounded solution of the previous system when
 $R$
tends to infinity. Furthermore, by imposing some restrictions on the powers
 $p,q,r,s$
without additional hypotheses on the weights
 $\rho_{i}$
, we obtain a second solution using variational methods. In this context we consider two particular cases: a gradient system and a Hamiltonian system.
Citation: Juan Arratia, Denilson Pereira, Pedro Ubilla. Elliptic systems involving Schrödinger operators with vanishing potentials. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021156
##### References:
 [1] A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar [2] H. Berestycki and P. L. Lions, Nonlinear scalar fields equation I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar [3] M.-F. Bidaut-Véron, Local behaviour of the solutions of a class of nonlinear elliptic systems, Adv. Differential Equations, 5 (2000), 147-192.   Google Scholar [4] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.  Google Scholar [5] J. A. Cardoso, P. Cerda, D. S. Pereira and P. Ubilla, Schrödinger equation with vanishing potentials involving Brezis-Kamin type problems, Discrete Contin. Dyn. Syst., 41 (2021), 2947-2969.  doi: 10.3934/dcds.2020392.  Google Scholar [6] B. D. Esry, C. H. Greene, J. P. Burke junior and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594.  Google Scholar [7] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.  Google Scholar [8] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472.   Google Scholar [9] G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.  Google Scholar [10] G. Li and H. Ye, Existence of positive solutions to semilinear elliptic systems in $\mathbb{R}^N$ with zero mass, Acta Math. Sci. Ser. B, 33 (2013), 913-928.  doi: 10.1016/S0252-9602(13)60050-8.  Google Scholar [11] C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron, 23 (1987), 174-176.  doi: 10.1109/JQE.1987.1073308.  Google Scholar [12] E. Mitidieri and S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N$, Proc. Steklov Inst. Math., 277 (1999), 1-32.   Google Scholar [13] M. Montenegro, The construction of principal spectral curves for Lane-Emden systems and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 193-229.   Google Scholar [14] P. Quittner, A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities, Nonlinear Anal., 102 (2014), 144-158.  doi: 10.1016/j.na.2014.02.010.  Google Scholar [15] P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.  doi: 10.1137/11085428X.  Google Scholar [16] M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258.   Google Scholar [17] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar [18] C. A. Stuart, An introduction to elliptic equations on $\mathbb{R}^N$, Nonlinear Funct. Anal. Appl. Diff. Eqs, World Sci., (1988), 237-285.  Google Scholar [19] E. Toon and P. Ubilla, Hamiltonian systems of Schrödinger equations with vanishing potentials, Commun. Contemp. Math., 2020. Google Scholar [20] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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##### References:
 [1] A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc., 7 (2005), 117-144.  doi: 10.4171/JEMS/24.  Google Scholar [2] H. Berestycki and P. L. Lions, Nonlinear scalar fields equation I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar [3] M.-F. Bidaut-Véron, Local behaviour of the solutions of a class of nonlinear elliptic systems, Adv. Differential Equations, 5 (2000), 147-192.   Google Scholar [4] H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.  Google Scholar [5] J. A. Cardoso, P. Cerda, D. S. Pereira and P. Ubilla, Schrödinger equation with vanishing potentials involving Brezis-Kamin type problems, Discrete Contin. Dyn. Syst., 41 (2021), 2947-2969.  doi: 10.3934/dcds.2020392.  Google Scholar [6] B. D. Esry, C. H. Greene, J. P. Burke junior and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594.  Google Scholar [7] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, 224, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.  Google Scholar [8] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472.   Google Scholar [9] G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.  Google Scholar [10] G. Li and H. Ye, Existence of positive solutions to semilinear elliptic systems in $\mathbb{R}^N$ with zero mass, Acta Math. Sci. Ser. B, 33 (2013), 913-928.  doi: 10.1016/S0252-9602(13)60050-8.  Google Scholar [11] C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron, 23 (1987), 174-176.  doi: 10.1109/JQE.1987.1073308.  Google Scholar [12] E. Mitidieri and S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems on $\mathbb{R}^N$, Proc. Steklov Inst. Math., 277 (1999), 1-32.   Google Scholar [13] M. Montenegro, The construction of principal spectral curves for Lane-Emden systems and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 (2000), 193-229.   Google Scholar [14] P. Quittner, A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities, Nonlinear Anal., 102 (2014), 144-158.  doi: 10.1016/j.na.2014.02.010.  Google Scholar [15] P. Quittner and P. Souplet, Symmetry of components for semilinear elliptic systems, SIAM J. Math. Anal., 44 (2012), 2545-2559.  doi: 10.1137/11085428X.  Google Scholar [16] M. A. S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations, 8 (1995), 1245-1258.   Google Scholar [17] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar [18] C. A. Stuart, An introduction to elliptic equations on $\mathbb{R}^N$, Nonlinear Funct. Anal. Appl. Diff. Eqs, World Sci., (1988), 237-285.  Google Scholar [19] E. Toon and P. Ubilla, Hamiltonian systems of Schrödinger equations with vanishing potentials, Commun. Contemp. Math., 2020. Google Scholar [20] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar
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