October  2019, 39(10): 5775-5784. doi: 10.3934/dcds.2019253

On a resonant and superlinear elliptic system

1. 

Universidade Federal do Espírito Santo, Departamento de Matemática, 29500-000, Alegre - ES, Brazil

2. 

Universidade Federal de São Carlos, Departamento de Matemática, 13565-905, São Carlos - SP, Brazil

Received  August 2018 Published  July 2019

Fund Project: The first author is supported by CAPES. The second author was supported by FAPESP grant 17/16108-6

We prove existence of solutions for a class of nonhomogeneous elliptic system with asymmetric nonlinearities that are resonant at −∞ and superlinear at +∞. The proof is based on topological degree arguments. A priori bounds for the solutions are obtained by adapting the method of BrezisTurner.

Citation: Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253
References:
[1]

H. Brezis and R. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations, 2 (1977), 601-614. doi: 10.1080/03605307708820041. Google Scholar

[2]

K. C. Chang, An extension of the Hess-Kato theorem to elliptic systems and its applications to multiple solutions problems,, Acta Math. Sin. (Eng. Ser.), 15 (1999), 439-454. doi: 10.1007/s10114-999-0078-0. Google Scholar

[3]

D. G. Costa and C. A. Magalhães, A variational approach to subquadratic perturbations of elliptic systems,, J. Differential Equations, 111 (1994), 103-122. doi: 10.1006/jdeq.1994.1077. Google Scholar

[4]

M. Cuesta and C. De Coster, Superlinear critical resonant problems with small forcing term, Cal. Var. Partial Differential Equations, 54 (2015), 349-363. doi: 10.1007/s00526-014-0788-8. Google Scholar

[5]

M. Cuesta and C. De Coster, A resonant-superlinear elliptic problem revisited,, Adv. Nonlinear Stud., 13 (2013), 97-114. doi: 10.1515/ans-2013-0106. Google Scholar

[6]

M. CuestaD. G. de Figueiredo and P. N. Srikanth, On a resonant superlinear elliptic problem, Calc. Var. Partial Differential Equations, 17 (2003), 221-233. Google Scholar

[7]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Lecture Notes in Math., 957 (1982), 34-87. Google Scholar

[8]

D. G. de Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Comm. Partial Differential Equations, 17 (1992), 339-346. doi: 10.1080/03605309208820844. Google Scholar

[9]

F. O. de Paiva and W. Rosa, Neumann problems with resonance in the first eigenvalue,, Math. Nachr., 290 (2017), 2198-2206. doi: 10.1002/mana.201600139. Google Scholar

[10]

M. F. Furtado and F. O. de Paiva, Multiplicity of solutions for resonant elliptic systems,, J. Math. Anal. Appl., 319 (2006), 435-449. doi: 10.1016/j.jmaa.2005.06.038. Google Scholar

[11]

R. Kannan and R. Ortega, Landesman-Lazer conditions for problems with ``one-side unbounded" nonlinearities,, Nonlinear. Anal., 9 (1985), 1313-1317. doi: 10.1016/0362-546X(85)90090-2. Google Scholar

[12]

R. Kannan and R. Ortega, Superlinear elliptic boundary value problems,, Czechoslovak Math. J., 37 (1987), 386-399. Google Scholar

[13]

J. R. Ward, Perturbations with some superlinear growth for a class of second order elliptic boundary value problems, Nonlinear Anal., 6 (1982), 367-374. doi: 10.1016/0362-546X(82)90022-0. Google Scholar

show all references

References:
[1]

H. Brezis and R. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations, 2 (1977), 601-614. doi: 10.1080/03605307708820041. Google Scholar

[2]

K. C. Chang, An extension of the Hess-Kato theorem to elliptic systems and its applications to multiple solutions problems,, Acta Math. Sin. (Eng. Ser.), 15 (1999), 439-454. doi: 10.1007/s10114-999-0078-0. Google Scholar

[3]

D. G. Costa and C. A. Magalhães, A variational approach to subquadratic perturbations of elliptic systems,, J. Differential Equations, 111 (1994), 103-122. doi: 10.1006/jdeq.1994.1077. Google Scholar

[4]

M. Cuesta and C. De Coster, Superlinear critical resonant problems with small forcing term, Cal. Var. Partial Differential Equations, 54 (2015), 349-363. doi: 10.1007/s00526-014-0788-8. Google Scholar

[5]

M. Cuesta and C. De Coster, A resonant-superlinear elliptic problem revisited,, Adv. Nonlinear Stud., 13 (2013), 97-114. doi: 10.1515/ans-2013-0106. Google Scholar

[6]

M. CuestaD. G. de Figueiredo and P. N. Srikanth, On a resonant superlinear elliptic problem, Calc. Var. Partial Differential Equations, 17 (2003), 221-233. Google Scholar

[7]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, Lecture Notes in Math., 957 (1982), 34-87. Google Scholar

[8]

D. G. de Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation,, Comm. Partial Differential Equations, 17 (1992), 339-346. doi: 10.1080/03605309208820844. Google Scholar

[9]

F. O. de Paiva and W. Rosa, Neumann problems with resonance in the first eigenvalue,, Math. Nachr., 290 (2017), 2198-2206. doi: 10.1002/mana.201600139. Google Scholar

[10]

M. F. Furtado and F. O. de Paiva, Multiplicity of solutions for resonant elliptic systems,, J. Math. Anal. Appl., 319 (2006), 435-449. doi: 10.1016/j.jmaa.2005.06.038. Google Scholar

[11]

R. Kannan and R. Ortega, Landesman-Lazer conditions for problems with ``one-side unbounded" nonlinearities,, Nonlinear. Anal., 9 (1985), 1313-1317. doi: 10.1016/0362-546X(85)90090-2. Google Scholar

[12]

R. Kannan and R. Ortega, Superlinear elliptic boundary value problems,, Czechoslovak Math. J., 37 (1987), 386-399. Google Scholar

[13]

J. R. Ward, Perturbations with some superlinear growth for a class of second order elliptic boundary value problems, Nonlinear Anal., 6 (1982), 367-374. doi: 10.1016/0362-546X(82)90022-0. Google Scholar

[1]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

[2]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499

[3]

Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145

[4]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731

[5]

Alfonso Castro, Rosa Pardo. A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 783-790. doi: 10.3934/dcdsb.2017038

[6]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258

[7]

Feng Du, Adriano Cavalcante Bezerra. Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting laplacian. Communications on Pure & Applied Analysis, 2017, 6 (2) : 475-491. doi: 10.3934/cpaa.2017024

[8]

Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277

[9]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303

[10]

Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721

[11]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013

[12]

Radjesvarane Alexandre, Jie Liao, Chunjin Lin. Some a priori estimates for the homogeneous Landau equation with soft potentials. Kinetic & Related Models, 2015, 8 (4) : 617-650. doi: 10.3934/krm.2015.8.617

[13]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765

[14]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861

[15]

Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971

[16]

Francesco Della Pietra, Ireneo Peral. Breaking of resonance for elliptic problems with strong degeneration at infinity. Communications on Pure & Applied Analysis, 2011, 10 (2) : 593-612. doi: 10.3934/cpaa.2011.10.593

[17]

Jiabao Su, Zhaoli Liu. A bounded resonance problem for semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 431-445. doi: 10.3934/dcds.2007.19.431

[18]

Patrick Winkert, Rico Zacher. A priori bounds for weak solutions to elliptic equations with nonstandard growth. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 865-878. doi: 10.3934/dcdss.2012.5.865

[19]

Tong Li, Hailiang Liu. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 511-521. doi: 10.3934/dcds.2009.24.511

[20]

Mei Ming. Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6039-6067. doi: 10.3934/dcds.2019264

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (56)
  • HTML views (88)
  • Cited by (0)

[Back to Top]