June  2021, 41(6): 2947-2969. doi: 10.3934/dcds.2020392

Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems

1. 

Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão-SE, 49100-000, Brazil

2. 

Departamento de Matematica y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

3. 

Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil

* Corresponding author

Received  December 2019 Revised  October 2020 Published  June 2021 Early access  December 2020

Fund Project: The first author is partially supported by FAPITEC/CAPES and by CNPq - Universal.
The second author was partially supported by Proyecto código 042033CL, Dirección de Investigación, Científica y Tecnológica, DICYT.
The third author was partially supported by Proyecto código 041933UL POSTDOC, Dirección de Investigación, Científica y Tecnológica, DICYT.
The fourth author was partially supported by FONDECYT grant 1181125, 1161635, 1171691

We prove the existence of a bounded positive solution for the following stationary Schrödinger equation
$ \begin{equation*} -\Delta u+V(x)u = f(x,u),\,\,\, x\in\mathbb{R}^n,\,\, n\geq 3, \end{equation*} $
where
$ V $
is a vanishing potential and
$ f $
has a sublinear growth at the origin (for example if
$ f(x,u) $
is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [6]. In addition, if
$ f $
has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type
$ \rho(x)f(u) $
where
$ f $
is a concave-convex function and
$ \rho $
satisfies the
$ \mathrm{(H)} $
property introduced in [6]. We also note that we do not impose any integrability assumptions on the function
$ \rho $
, which is imposed in most works.
Citation: Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392
References:
[1]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[2]

A. AmbrosettiV. 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.

[3]

A. BahrouniH. Ounaies and V. D. Rădulescu, Bound state solutions of sublinear Schrödinger equations with lack of compactness, RACSAM, 113 (2019), 1191-1210.  doi: 10.1007/s13398-018-0541-9.

[4]

A. BahrouniH. Ounaies and V. D. Rădulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.  doi: 10.1017/S0308210513001169.

[5]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications., 1 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.

[6]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[7]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[8]

J. Chabrowski and J. M. B. do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.  doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R.

[9]

D. G. de FigueiredoJ-P Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.

[10]

D. G. de FigueiredoJ-P Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286.  doi: 10.4171/JEMS/52.

[11]

F. Gazzola and A. Malchiodi, Some remark on the equation $-\Delta u = \lambda(1+u)^p$ for varying $\lambda, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845.  doi: 10.1081/PDE-120002875.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.

[13]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math., vol. 1, AMS, Providence, RI, 1997.

[14]

T-S Hsu and H-L Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\mathbb{R}^n$, J. Math. Anal. Appl., 365 (2010), 758-775.  doi: 10.1016/j.jmaa.2009.12.004.

[15]

Z. Liu and Z-Q Wang, Schrödinger equations with concave and convex nonlinearities, Z. angew. Math. Phys., 56 (2005), 609-629.  doi: 10.1007/s00033-005-3115-6.

[16]

M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice Hall, Englewoood Cliffs, New Jersey, 1967.

[17]

T-F Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^n$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.

show all references

References:
[1]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[2]

A. AmbrosettiV. 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.

[3]

A. BahrouniH. Ounaies and V. D. Rădulescu, Bound state solutions of sublinear Schrödinger equations with lack of compactness, RACSAM, 113 (2019), 1191-1210.  doi: 10.1007/s13398-018-0541-9.

[4]

A. BahrouniH. Ounaies and V. D. Rădulescu, Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 445-465.  doi: 10.1017/S0308210513001169.

[5]

H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis. Theory, Methods & Applications., 1 (1986), 55-64.  doi: 10.1016/0362-546X(86)90011-8.

[6]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N$, Manuscripta Math., 74 (1992), 87-106.  doi: 10.1007/BF02567660.

[7]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[8]

J. Chabrowski and J. M. B. do Ó, On semilinear elliptic equations involving concave and convex nonlinearities, Math. Nachr., 233/234 (2002), 55-76.  doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.0.CO;2-R.

[9]

D. G. de FigueiredoJ-P Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.

[10]

D. G. de FigueiredoJ-P Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc., 8 (2006), 269-286.  doi: 10.4171/JEMS/52.

[11]

F. Gazzola and A. Malchiodi, Some remark on the equation $-\Delta u = \lambda(1+u)^p$ for varying $\lambda, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845.  doi: 10.1081/PDE-120002875.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0.

[13]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lect. Notes Math., vol. 1, AMS, Providence, RI, 1997.

[14]

T-S Hsu and H-L Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\mathbb{R}^n$, J. Math. Anal. Appl., 365 (2010), 758-775.  doi: 10.1016/j.jmaa.2009.12.004.

[15]

Z. Liu and Z-Q Wang, Schrödinger equations with concave and convex nonlinearities, Z. angew. Math. Phys., 56 (2005), 609-629.  doi: 10.1007/s00033-005-3115-6.

[16]

M. H. Protter and H. F. Weinberger, Maximum Principle in Differential Equations, Prentice Hall, Englewoood Cliffs, New Jersey, 1967.

[17]

T-F Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^n$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131.  doi: 10.1016/j.jfa.2009.08.005.

[1]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076

[2]

Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427

[3]

Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107

[4]

Junping Shi, Ratnasingham Shivaji. Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 559-571. doi: 10.3934/dcds.2001.7.559

[5]

Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575

[6]

João Marcos do Ó, Uberlandio Severo. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Communications on Pure and Applied Analysis, 2009, 8 (2) : 621-644. doi: 10.3934/cpaa.2009.8.621

[7]

M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3445-3479. doi: 10.3934/cpaa.2021113

[8]

Leszek Gasiński, Nikolaos S. Papageorgiou. Singular equations with variable exponents and concave-convex nonlinearities. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022135

[9]

Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure and Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815

[10]

Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2089-2104. doi: 10.3934/cpaa.2017103

[11]

Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315

[12]

Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073

[13]

Jianqing Chen. A variational argument to finding global solutions of a quasilinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (1) : 83-88. doi: 10.3934/cpaa.2008.7.83

[14]

Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008

[15]

Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036

[16]

Chunyan Ji, Yang Xue, Yong Li. Periodic solutions for SDEs through upper and lower solutions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4737-4754. doi: 10.3934/dcdsb.2020122

[17]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715

[18]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[19]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

[20]

João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (293)
  • HTML views (164)
  • Cited by (0)

[Back to Top]