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 and Applied Analysis, 2020, 19 (9) : 4401-4432. doi: 10.3934/cpaa.2020201
References:
[1] R. A. Adams and J. F. Fournier, Sobolev spaces, 2$^{nd}$ edition, Academic Press, New York, 2003. 
[2]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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. 

[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. 

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[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.

[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.

show all references

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

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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. 

[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. 

[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.

[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.

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[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.

[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.

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