September  2020, 19(9): 4401-4432. doi: 10.3934/cpaa.2020201

Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials

1. 

Universidade Federal de Goiás, IME, Goiânia-GO, Brazil

2. 

Universidade Federal de Jataí, Jataí-GO, Brazil

3. 

Universidade de Brasília, Brasília-DF, Brazil

* Corresponding author

Received  October 2019 Revised  March 2020 Published  June 2020

Fund Project: The second author was partially supported by CNPq grants 429955/2018-9

It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by $ (\Phi_{1}, \Phi_{2}) $-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases.

Citation: Marcos L. M. Carvalho, Edcarlos D. Silva, Claudiney Goulart, Carlos A. Santos. Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4401-4432. doi: 10.3934/cpaa.2020201
References:
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S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var., 1 (1993), 439-475.  doi: 10.1007/BF01206962.  Google Scholar

[3]

S. Alama and M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. Henri Poincare Anal. Non Lineaire, 13 (1996), 95-115.  doi: 10.1016/S0294-1449(16)30098-1.  Google Scholar

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C. O. AlvesF. J. S. A. Corrêa and J. V. A. Gonçalves, Existence of solutions for some classes of singular Hamiltonian systems, Adv. Nonlinear Stud., 5 (2005), 265-278.  doi: 10.1515/ans-2005-0206.  Google Scholar

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G. BonannoG. Molica Bisci and V. Rădulescu, Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces, Nonlinear Anal. Theory Meth. Appl., 75 (2012), 4441-4456.  doi: 10.1016/j.na.2011.12.016.  Google Scholar

[7]

K. J. Brown and Y. Zhang, The Nehari manifold for semilinear elliptic equation with a sign-changing weight function, J. Differ. Equ., 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[8]

K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differ. Equ., 69 (2007), 1-9.   Google Scholar

[9]

M. L. CarvalhoO. H. Myagaki and C. Goulart, Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth, Commun. Pure Appl. Anal., 18 (2019), 83-106.  doi: 10.3934/cpaa.2019006.  Google Scholar

[10]

F. J. S. A. CorrêaM. L. M. CarvalhoJ. V. Goncalves and E. D. Silva, Sign changing solutions for quasilinear superlinear elliptic problems, Quart. J. Math., 68 (2017), 391-420.  doi: 10.1093/qmath/haw047.  Google Scholar

[11]

E. D. da SilvaM. L. CarvalhoJ. V. Goncalves and C. Goulart, Critical quasilinear elliptic problems using concave-convex nonlinearities, Ann. Mat. Pura Appl., 198 (2019), 693-726.  doi: 10.1007/s10231-018-0794-0.  Google Scholar

[12]

F. O. V. de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity, J. Funct. Anal., 261 (2011), 2569-2586.  doi: 10.1016/j.jfa.2011.07.002.  Google Scholar

[13]

E. DiBenedetto, $C^{1, \gamma}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1985), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[14]

T. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and embedding theorems, J. Funct. Anal., 8 (1971), 52-75.  doi: 10.1016/0022-1236(71)90018-8.  Google Scholar

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[17]

S. El ManouniK. Perera and R. Shivaji, On singular quasimonotone $(p, q)$-Laplacian systems, Proc. R. Soc. Edinb. Sect. A, 142 (2012), 585-594.  doi: 10.1017/S0308210510001356.  Google Scholar

[18]

H. Fan, Multiple positive solutions for semi-linear elliptic systems with sign-changing weight, J. Math. Anal. Appl., 409 (2014), 399-408.  doi: 10.1016/j.jmaa.2013.07.014.  Google Scholar

[19]

N. Fukagai and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}^N$, Funkc. Ekvacioj, 49 (2006), 235-267.  doi: 10.1619/fesi.49.235.  Google Scholar

[20]

N. Fukagai and K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat., 186 (2007), 539-564.  doi: 10.1007/s10231-006-0018-x.  Google Scholar

[21]

J. García-Azorero and I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  doi: 10.2307/2001562.  Google Scholar

[22]

J. GiacomoniI. Schindler and P. Takác, Singular quasilinear elliptic systems and H$\ddot{o}$lder regularity, Adv. Differ. Equ., 20 (2015), 259-298.   Google Scholar

[23]

J. V. Goncalves and M. L. Carvalho, Multivalued equations on a bounded domain via minimization on Orlicz-Sobolev spaces, J. Convex Anal., 21 (2014), 201-218.   Google Scholar

[24]

J. V. GoncalvesM. L. Carvalho and C. A. Santos, Quasilinear elliptic systems with convex-concave singular terms $\Phi$-Laplacian operator, Differ. Integral Equ., 31 (2018), 231-256.   Google Scholar

[25]

J. V. GoncalvesM. L. Carvalho and C. A. Santos, About positive $W_loc^{1, \Phi}(\Omega)$-solutions to quasilinear elliptic problems with singular semilinear term, Topol. Meth. Nonlinear Anal., 53 (2019), 491-517.  doi: 10.12775/tmna.2019.009.  Google Scholar

[26]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.  doi: 10.2307/1996957.  Google Scholar

[27]

J. P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, Nonlinear Anal. Funct. Spaces Appl., 26 (2013), 59-94.   Google Scholar

[28]

D. D. Hai, Singular elliptic systems with asymptotically linear nonlinearities, Differ. Integral Equ., 190 (1978), 837-844.   Google Scholar

[29]

T. S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities, Nonlinear Anal., 71 (2009), 2688-2698.  doi: 10.1016/j.na.2009.01.110.  Google Scholar

[30]

T. S. Hsu, Multiple positive solutions for a quasilinear elliptic system involving concave-convex nonlinearities and Sign-Changing Weight Functions, Int. J. Math. Math. Sci., (2012), Art. 109214. doi: 10.1155/2012/109214.  Google Scholar

[31]

J. Huentutripay and R. Manasevich, Nonlinear eigenvalues for a quasilinear elliptic system in Orlicz–Sobolev spaces, J. Dyn. Differ. Equ., 18 (2006), 901-929.  doi: 10.1007/s10884-006-9049-7.  Google Scholar

[32]

Q. Lia and Z. Yang, Multiplicity of positive solutions for a $(p, q)$-Laplacian system with concave and critical nonlinearities, J. Math. Anal. Appl., 423 (2015), 660-680.  doi: 10.1016/j.jmaa.2014.10.009.  Google Scholar

[33]

M. Mihăilescu and V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl., 330 (2007), 416-432.  doi: 10.1016/j.jmaa.2006.07.082.  Google Scholar

[34]

M. RamosS. Terracini and C. Troestler, Superlinear indefinite elliptic problems and Pohožaev type identities, J. Funct. Anal., 159 (1998), 596-628.  doi: 10.1006/jfan.1998.3332.  Google Scholar

[35]

M. N. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.  Google Scholar

[36]

Z. Tan and F. Fang, Orlicz-Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 402 (2013), 348-370.  doi: 10.1016/j.jmaa.2013.01.029.  Google Scholar

[37]

G. Tarantello, On nonhomogenous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincare Anal. Non Lineaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar

[38]

G. Tarantello, Multiplicity results for an inhomogeneous Neumann problem with critical exponent, Manuscr. Math., 81 (1993), 57-78.  doi: 10.1007/BF02567844.  Google Scholar

[39]

L. YijingW. Shaoping and L. Yiming, Combined effects of singular and superlinear nonlinearities in Some singular boundary value problems, J. Differ. Equ., 176 (2001), 511-531.  doi: 10.1006/jdeq.2000.3973.  Google Scholar

[40]

S. Yijing and W. Shaoping, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257-1284.  doi: 10.1016/j.jfa.2010.11.018.  Google Scholar

[41]

S. Yijing and L. Shujie, Some remarks on a superlinear-singular problem: Estimates of $\lambda^{*}$, Nonlinear Anal., 69 (2008), 2636-2650.  doi: 10.1016/j.na.2007.08.037.  Google Scholar

[42]

T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270.  doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar

[43]

T. F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal., 68 (2008), 1733-1745.  doi: 10.1016/j.na.2007.01.004.  Google Scholar

[44]

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.  Google Scholar

show all references

References:
[1] R. A. Adams and J. F. Fournier, Sobolev spaces, 2$^{nd}$ edition, Academic Press, New York, 2003.   Google Scholar
[2]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var., 1 (1993), 439-475.  doi: 10.1007/BF01206962.  Google Scholar

[3]

S. Alama and M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Ann. Inst. Henri Poincare Anal. Non Lineaire, 13 (1996), 95-115.  doi: 10.1016/S0294-1449(16)30098-1.  Google Scholar

[4]

C. O. AlvesF. J. S. A. Corrêa and J. V. A. Gonçalves, Existence of solutions for some classes of singular Hamiltonian systems, Adv. Nonlinear Stud., 5 (2005), 265-278.  doi: 10.1515/ans-2005-0206.  Google Scholar

[5]

G. BonannoG. Molica Bisci and V. Rădulescu, Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces, Nonlinear Anal. Theory Meth. Appl., 74 (2011), 4785-4795.  doi: 10.1016/j.na.2011.04.049.  Google Scholar

[6]

G. BonannoG. Molica Bisci and V. Rădulescu, Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces, Nonlinear Anal. Theory Meth. Appl., 75 (2012), 4441-4456.  doi: 10.1016/j.na.2011.12.016.  Google Scholar

[7]

K. J. Brown and Y. Zhang, The Nehari manifold for semilinear elliptic equation with a sign-changing weight function, J. Differ. Equ., 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[8]

K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differ. Equ., 69 (2007), 1-9.   Google Scholar

[9]

M. L. CarvalhoO. H. Myagaki and C. Goulart, Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth, Commun. Pure Appl. Anal., 18 (2019), 83-106.  doi: 10.3934/cpaa.2019006.  Google Scholar

[10]

F. J. S. A. CorrêaM. L. M. CarvalhoJ. V. Goncalves and E. D. Silva, Sign changing solutions for quasilinear superlinear elliptic problems, Quart. J. Math., 68 (2017), 391-420.  doi: 10.1093/qmath/haw047.  Google Scholar

[11]

E. D. da SilvaM. L. CarvalhoJ. V. Goncalves and C. Goulart, Critical quasilinear elliptic problems using concave-convex nonlinearities, Ann. Mat. Pura Appl., 198 (2019), 693-726.  doi: 10.1007/s10231-018-0794-0.  Google Scholar

[12]

F. O. V. de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity, J. Funct. Anal., 261 (2011), 2569-2586.  doi: 10.1016/j.jfa.2011.07.002.  Google Scholar

[13]

E. DiBenedetto, $C^{1, \gamma}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1985), 827-850.  doi: 10.1016/0362-546X(83)90061-5.  Google Scholar

[14]

T. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and embedding theorems, J. Funct. Anal., 8 (1971), 52-75.  doi: 10.1016/0022-1236(71)90018-8.  Google Scholar

[15]

P. Drabek and J. Milota, Mehtods of Nonlinear Analysis, 2$^{nd}$ edition, Birkhaser Advanced Texts, New York, 2013. doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[16]

P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proc. R. Soc. Edinb. Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.  Google Scholar

[17]

S. El ManouniK. Perera and R. Shivaji, On singular quasimonotone $(p, q)$-Laplacian systems, Proc. R. Soc. Edinb. Sect. A, 142 (2012), 585-594.  doi: 10.1017/S0308210510001356.  Google Scholar

[18]

H. Fan, Multiple positive solutions for semi-linear elliptic systems with sign-changing weight, J. Math. Anal. Appl., 409 (2014), 399-408.  doi: 10.1016/j.jmaa.2013.07.014.  Google Scholar

[19]

N. Fukagai and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}^N$, Funkc. Ekvacioj, 49 (2006), 235-267.  doi: 10.1619/fesi.49.235.  Google Scholar

[20]

N. Fukagai and K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat., 186 (2007), 539-564.  doi: 10.1007/s10231-006-0018-x.  Google Scholar

[21]

J. García-Azorero and I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  doi: 10.2307/2001562.  Google Scholar

[22]

J. GiacomoniI. Schindler and P. Takác, Singular quasilinear elliptic systems and H$\ddot{o}$lder regularity, Adv. Differ. Equ., 20 (2015), 259-298.   Google Scholar

[23]

J. V. Goncalves and M. L. Carvalho, Multivalued equations on a bounded domain via minimization on Orlicz-Sobolev spaces, J. Convex Anal., 21 (2014), 201-218.   Google Scholar

[24]

J. V. GoncalvesM. L. Carvalho and C. A. Santos, Quasilinear elliptic systems with convex-concave singular terms $\Phi$-Laplacian operator, Differ. Integral Equ., 31 (2018), 231-256.   Google Scholar

[25]

J. V. GoncalvesM. L. Carvalho and C. A. Santos, About positive $W_loc^{1, \Phi}(\Omega)$-solutions to quasilinear elliptic problems with singular semilinear term, Topol. Meth. Nonlinear Anal., 53 (2019), 491-517.  doi: 10.12775/tmna.2019.009.  Google Scholar

[26]

J. P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190 (1974), 163-205.  doi: 10.2307/1996957.  Google Scholar

[27]

J. P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, Nonlinear Anal. Funct. Spaces Appl., 26 (2013), 59-94.   Google Scholar

[28]

D. D. Hai, Singular elliptic systems with asymptotically linear nonlinearities, Differ. Integral Equ., 190 (1978), 837-844.   Google Scholar

[29]

T. S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities, Nonlinear Anal., 71 (2009), 2688-2698.  doi: 10.1016/j.na.2009.01.110.  Google Scholar

[30]

T. S. Hsu, Multiple positive solutions for a quasilinear elliptic system involving concave-convex nonlinearities and Sign-Changing Weight Functions, Int. J. Math. Math. Sci., (2012), Art. 109214. doi: 10.1155/2012/109214.  Google Scholar

[31]

J. Huentutripay and R. Manasevich, Nonlinear eigenvalues for a quasilinear elliptic system in Orlicz–Sobolev spaces, J. Dyn. Differ. Equ., 18 (2006), 901-929.  doi: 10.1007/s10884-006-9049-7.  Google Scholar

[32]

Q. Lia and Z. Yang, Multiplicity of positive solutions for a $(p, q)$-Laplacian system with concave and critical nonlinearities, J. Math. Anal. Appl., 423 (2015), 660-680.  doi: 10.1016/j.jmaa.2014.10.009.  Google Scholar

[33]

M. Mihăilescu and V. Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl., 330 (2007), 416-432.  doi: 10.1016/j.jmaa.2006.07.082.  Google Scholar

[34]

M. RamosS. Terracini and C. Troestler, Superlinear indefinite elliptic problems and Pohožaev type identities, J. Funct. Anal., 159 (1998), 596-628.  doi: 10.1006/jfan.1998.3332.  Google Scholar

[35]

M. N. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.  Google Scholar

[36]

Z. Tan and F. Fang, Orlicz-Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 402 (2013), 348-370.  doi: 10.1016/j.jmaa.2013.01.029.  Google Scholar

[37]

G. Tarantello, On nonhomogenous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincare Anal. Non Lineaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar

[38]

G. Tarantello, Multiplicity results for an inhomogeneous Neumann problem with critical exponent, Manuscr. Math., 81 (1993), 57-78.  doi: 10.1007/BF02567844.  Google Scholar

[39]

L. YijingW. Shaoping and L. Yiming, Combined effects of singular and superlinear nonlinearities in Some singular boundary value problems, J. Differ. Equ., 176 (2001), 511-531.  doi: 10.1006/jdeq.2000.3973.  Google Scholar

[40]

S. Yijing and W. Shaoping, An exact estimate result for a class of singular equations with critical exponents, J. Funct. Anal., 260 (2011), 1257-1284.  doi: 10.1016/j.jfa.2010.11.018.  Google Scholar

[41]

S. Yijing and L. Shujie, Some remarks on a superlinear-singular problem: Estimates of $\lambda^{*}$, Nonlinear Anal., 69 (2008), 2636-2650.  doi: 10.1016/j.na.2007.08.037.  Google Scholar

[42]

T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270.  doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar

[43]

T. F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal., 68 (2008), 1733-1745.  doi: 10.1016/j.na.2007.01.004.  Google Scholar

[44]

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.  Google Scholar

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Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021021

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