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July  2022, 42(7): 3329-3353. doi: 10.3934/dcds.2022017

Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz condition

1. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro-UFRJ, Rio de Janeiro, RJ 21945-970, Brazil

2. 

Departamento de Matemática, Universidade de Brasília-UnB, Brasília, DF 70910-900, Brazil

3. 

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

* Corresponding author: Elard J. Hurtado

Received  August 2021 Revised  December 2021 Published  July 2022 Early access  March 2022

In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations $( \mathscr{P}_\lambda)$ in a smooth bounded domain, driven by a nonlocal integrodifferential operator $ \mathscr{L}_{\mathcal{A}K} $ with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem $( \mathscr{P}_\lambda)$ and we show that the problem treated has at least one nontrivial solution for any parameter $ \lambda >0 $ small enough as well as that the solution blows up, in the fractional Sobolev norm, as $ \lambda \to 0 $. Moreover, for the sublinear case, by imposing some additional hypotheses on the nonlinearity $ f(x,\cdot) $, and by using a new version of the symmetric Mountain Pass Theorem due to Kajikiya [18], we obtain the existence of infinitely many weak solutions which tend to zero, in the fractional Sobolev norm, for any parameter $ \lambda >0 $. As far as we know, the results of this paper are new in the literature.

Citation: Lauren M. M. Bonaldo, Elard J. Hurtado, Olímpio H. Miyagaki. Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz condition. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3329-3353. doi: 10.3934/dcds.2022017
References:
[1]

C. O. Alves and M. C. Ferreira, Nonlinear perturbations of a $p(x)$-Laplacian equation with critical growth in $\mathbb{R}^{N}$, Math. Nachr., 287 (2014), 849-868.  doi: 10.1002/mana.201200336.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and apllications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.

[3]

A. Bahrouni and K. Ho, Remarks on eigenvalue problems for fractional $p(\cdot)$-Laplacian, Asymptotic Analysis, 123 (2021), 139-156.  doi: 10.3233/ASY-201628.

[4]

A. Bahrouni and V. D. Rǎdulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. S, 11 (2018), 379-389.  doi: 10.3934/dcdss.2018021.

[5]

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 11 (1995), 3555-3561.  doi: 10.1090/S0002-9939-1995-1301008-2.

[6] G. M. BisciV. D. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.
[7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. 
[8]

L. M. M. Bonaldo, E. J. Hurtado and O. H. Miyagaki, A class of elliptic equations involving nonlocal integrodifferential operators with sign-changing weight functions, Journal of Mathematical Physics, 61 (2020), 051503, 26 pp. doi: 10.1063/1.5145154.

[9]

L. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differential Equations, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3.

[10]

G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336. 

[11]

D. G. Costa and C. A. Magalhães, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal., 24 (1995), 409-418.  doi: 10.1016/0362-546X(94)E0046-J.

[12]

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, Vol. 2017, Springer-Verlag, Heidelberg, (2011). doi: 10.1007/978-3-642-18363-8.

[13]

M. Fabian, P. Habala, P Hájek, V. Montesinos and V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis, New York Springer, 2011. doi: 10.1007/978-1-4419-7515-7.

[14]

K. Ho and Y.-H. Kim, A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian, Nonlinear Analysis, 188 (2019), 179-201.  doi: 10.1016/j.na.2019.06.001.

[15]

E. J. Hurtado, Nonlocal diffusion equations involving the fractional $p(\cdot)$-Laplacian, Journal of Dynamics and Differential Equations, 32 (2020), 557-587.  doi: 10.1007/s10884-019-09745-2.

[16]

E. J. HurtadoO. H. Miyagaki and R. S. Rodrigues, Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions, Journal of Dynamics and Differential Equations, 30 (2018), 405-432.  doi: 10.1007/s10884-016-9542-6.

[17]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbb{R}^{N}$, Proc. Roy. Soc. Edinburgh A., 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.

[18]

R. Kajikiya, A critical point theorem related to the symmetric mountain-pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.  doi: 10.1016/j.jfa.2005.04.005.

[19]

U. KaufmannJ. D. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacian, Electron. J. Qual. Theory Differ. Equ., 76 (2017), 1-10.  doi: 10.14232/ejqtde.2017.1.76.

[20]

J. Lee and Y.-H. Kim, Multiplicity results for nonlinear Neumann boundary value problems involving $p$-Laplace type operators, Bound. Value Probl., 2016 (2016), Paper No. 95, 25 pp. doi: 10.1186/s13661-016-0603-x.

[21]

S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016.

[22]

R. Pei, Fractional $p$-laplacian equations with subcritical and critical exponential growth without the Ambrosett-Rabinowitz condition, Mediterranean Journal of Mathematics, 15 (2018), Paper No. 66, 15 pp. doi: 10.1007/s00009-018-1115-y.

[23]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. inMath. 65. American Mathematical Society, Providence, RI, (1986). doi: 10.1090/cbms/065.

[24]

B. ZhangG. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.  doi: 10.1088/0951-7715/28/7/2247.

[25]

Q.-M. Zhou, On a class of superlinear nonlocal fractional problems without Ambrosetti-Rabinowitz type conditions, Electronic Journal of Qualitative Theory of Differential Equations, (2019), Paper No. 17, 12 pp. doi: 10.14232/ejqtde.2019.1.17.

[26]

Q.-M. Zhou, On the superlinear problems involving $p(x)$-Laplacian-like operators without AR-condition, Nonlinear Analysis: Real World Applications, 21 (2015), 161-169.  doi: 10.1016/j.nonrwa.2014.07.003.

show all references

References:
[1]

C. O. Alves and M. C. Ferreira, Nonlinear perturbations of a $p(x)$-Laplacian equation with critical growth in $\mathbb{R}^{N}$, Math. Nachr., 287 (2014), 849-868.  doi: 10.1002/mana.201200336.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and apllications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.

[3]

A. Bahrouni and K. Ho, Remarks on eigenvalue problems for fractional $p(\cdot)$-Laplacian, Asymptotic Analysis, 123 (2021), 139-156.  doi: 10.3233/ASY-201628.

[4]

A. Bahrouni and V. D. Rǎdulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. S, 11 (2018), 379-389.  doi: 10.3934/dcdss.2018021.

[5]

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 11 (1995), 3555-3561.  doi: 10.1090/S0002-9939-1995-1301008-2.

[6] G. M. BisciV. D. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316282397.
[7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. 
[8]

L. M. M. Bonaldo, E. J. Hurtado and O. H. Miyagaki, A class of elliptic equations involving nonlocal integrodifferential operators with sign-changing weight functions, Journal of Mathematical Physics, 61 (2020), 051503, 26 pp. doi: 10.1063/1.5145154.

[9]

L. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differential Equations, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3.

[10]

G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A, 112 (1978), 332-336. 

[11]

D. G. Costa and C. A. Magalhães, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal., 24 (1995), 409-418.  doi: 10.1016/0362-546X(94)E0046-J.

[12]

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, Vol. 2017, Springer-Verlag, Heidelberg, (2011). doi: 10.1007/978-3-642-18363-8.

[13]

M. Fabian, P. Habala, P Hájek, V. Montesinos and V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis, New York Springer, 2011. doi: 10.1007/978-1-4419-7515-7.

[14]

K. Ho and Y.-H. Kim, A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian, Nonlinear Analysis, 188 (2019), 179-201.  doi: 10.1016/j.na.2019.06.001.

[15]

E. J. Hurtado, Nonlocal diffusion equations involving the fractional $p(\cdot)$-Laplacian, Journal of Dynamics and Differential Equations, 32 (2020), 557-587.  doi: 10.1007/s10884-019-09745-2.

[16]

E. J. HurtadoO. H. Miyagaki and R. S. Rodrigues, Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions, Journal of Dynamics and Differential Equations, 30 (2018), 405-432.  doi: 10.1007/s10884-016-9542-6.

[17]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbb{R}^{N}$, Proc. Roy. Soc. Edinburgh A., 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.

[18]

R. Kajikiya, A critical point theorem related to the symmetric mountain-pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.  doi: 10.1016/j.jfa.2005.04.005.

[19]

U. KaufmannJ. D. Rossi and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacian, Electron. J. Qual. Theory Differ. Equ., 76 (2017), 1-10.  doi: 10.14232/ejqtde.2017.1.76.

[20]

J. Lee and Y.-H. Kim, Multiplicity results for nonlinear Neumann boundary value problems involving $p$-Laplace type operators, Bound. Value Probl., 2016 (2016), Paper No. 95, 25 pp. doi: 10.1186/s13661-016-0603-x.

[21]

S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016.

[22]

R. Pei, Fractional $p$-laplacian equations with subcritical and critical exponential growth without the Ambrosett-Rabinowitz condition, Mediterranean Journal of Mathematics, 15 (2018), Paper No. 66, 15 pp. doi: 10.1007/s00009-018-1115-y.

[23]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. inMath. 65. American Mathematical Society, Providence, RI, (1986). doi: 10.1090/cbms/065.

[24]

B. ZhangG. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.  doi: 10.1088/0951-7715/28/7/2247.

[25]

Q.-M. Zhou, On a class of superlinear nonlocal fractional problems without Ambrosetti-Rabinowitz type conditions, Electronic Journal of Qualitative Theory of Differential Equations, (2019), Paper No. 17, 12 pp. doi: 10.14232/ejqtde.2019.1.17.

[26]

Q.-M. Zhou, On the superlinear problems involving $p(x)$-Laplacian-like operators without AR-condition, Nonlinear Analysis: Real World Applications, 21 (2015), 161-169.  doi: 10.1016/j.nonrwa.2014.07.003.

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