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

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