January 2019, 18(1): 83-106. doi: 10.3934/cpaa.2019006

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth

1. 

IME - Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-970-Goiania-GO, Brazil

2. 

R. Riachuelo, 1530 - Setor Samuel Graham, Universidade Federal de Goiás, 75804-020- Jataí-GO, Brazil

3. 

Departamento de Matemática, Universidade Federal de Juiz de Fora, 36036-330-Juiz de Fora-MG, Brazil

* Corresponding author

Received  August 2017 Revised  August 2017 Published  August 2018

Fund Project: O. H. Miyagaki is corresponding author and he received research grants from CNPq/Brazil and INCTMAT/CNPQ/Brazil

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by the Φ-Laplacian operator. One of the solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev space framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.

Citation: Marcos L. M. Carvalho, José Valdo A. Goncalves, Claudiney Goulart, Olímpio H. Miyagaki. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 83-106. doi: 10.3934/cpaa.2019006
References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces, Academic Press, New York, 2003.

[2]

H. Brézis and L. Nirenberg, A minimization problem with critical exponent and nonzero data, in Symmetry in nature, Quaderni di Scuola Normale Superiore, Pisa, pp. 129-140, 1989.

[3]

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

[4]

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

[5]

K. J. Brown and T.-F. Wu, A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, 22 (2009), 1097-1114.

[6]

D. M. CaoG. B. Li and H. S. Zhou, Multiple solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1177-1191. doi: 10.1017/S0308210500030183.

[7]

M. L. Carvalho, E. D. da Silva and C. Goulart, Quasilinear elliptic problems with concave-convex nonlinearities, Communications in Contemporary Mathematics, 19 (2017), 1650050, 25 pp. doi: 10.1142/S0219199716500504.

[8]

M. L. Carvalho, E. D. da Silva, J. V. A. Goncalves and C. Goulart, Concave-convex effects for critical quasilinear elliptic problems, preprint.

[9]

M. L. CarvalhoF. J. S. CorreaJ. V. A. Goncalves and E. D. da Silva, Sign changing solutions for quasilinear superlinear elliptic problems, Quarterly Journal of Mathematics, 68 (2017), 391-420. doi: 10.1093/qmath/haw047.

[10]

J. Chabrowski, On multiple solutions for the nonhomogeneous p-Laplacian with a critical Sobolev exponent, Differential Integral Equations, 8 (1995), 705-716.

[11]

M. ClappO. Kavian and B. Ruf, Multiple solutions of nonhomogeneous elliptic equations with critical nonlinearity on symmetric domains, Communications in Contemporary Mathematics, 5 (2003), 147-169. doi: 10.1142/S0219199703000963.

[12]

T. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis, 8 (1971), 52-75.

[13]

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

[14]

G. M. Figueiredo and H. Ramos Quoirin, Ground states of elliptic problems involving non-homogeneous operators, Indiana Univ. Math. J., 65 (2016), 779-795. doi: 10.1512/iumj.2016.65.5828.

[15]

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

[16]

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

[17]

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

[18]

J. P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, in Nonlinear Analysis, Function Spaces and Applications, Proc. Spring School, Horni Bradlo, 1978, Teubner, Leipzig, 59-94, 1979.

[19]

N. Hirano and N. Shioji, A multiplicity result including a sign-changing solution for an inhomogeneous Neumann problem with critical exponent, Proc. Royal Soc. Edinburgh Sect. A, 137 (2007), 333-347. doi: 10.1017/S0308210505001277.

[20]

D. Motreanu and M. Tanaka, Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter, Annali di Matematica Pura ed Applicata, 193 (2014), 1255-1282. doi: 10.1007/s10231-013-0327-9.

[21]

D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear (p, q)-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937. doi: 10.1090/S0002-9947-2013-06124-7.

[22]

M. N. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1985.

[23]

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.

[24]

M. Tanaka, Generalized eigenvalue problems for (p, q)-Laplacian with indefinite weight, J. Math. Anal. Appl., 419 (2014), 1181-1192. doi: 10.1016/j.jmaa.2014.05.044.

[25]

G. Tarantello, On nonhomogenous elliptic equations involving critical Sobolev expoent, I. H. Poincaré, Analyse Non Linéaire, 9 (1992), 281-304. doi: 10.1016/S0294-1449(16)30238-4.

[26]

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

[27]

M. Willem, Minimax Theorems, Birkhäuser Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[28]

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.

[29]

T.-F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^N$ involving sign-changing weight, J. Functional Analysis, 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, Academic Press, New York, 2003.

[2]

H. Brézis and L. Nirenberg, A minimization problem with critical exponent and nonzero data, in Symmetry in nature, Quaderni di Scuola Normale Superiore, Pisa, pp. 129-140, 1989.

[3]

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

[4]

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

[5]

K. J. Brown and T.-F. Wu, A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, 22 (2009), 1097-1114.

[6]

D. M. CaoG. B. Li and H. S. Zhou, Multiple solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1177-1191. doi: 10.1017/S0308210500030183.

[7]

M. L. Carvalho, E. D. da Silva and C. Goulart, Quasilinear elliptic problems with concave-convex nonlinearities, Communications in Contemporary Mathematics, 19 (2017), 1650050, 25 pp. doi: 10.1142/S0219199716500504.

[8]

M. L. Carvalho, E. D. da Silva, J. V. A. Goncalves and C. Goulart, Concave-convex effects for critical quasilinear elliptic problems, preprint.

[9]

M. L. CarvalhoF. J. S. CorreaJ. V. A. Goncalves and E. D. da Silva, Sign changing solutions for quasilinear superlinear elliptic problems, Quarterly Journal of Mathematics, 68 (2017), 391-420. doi: 10.1093/qmath/haw047.

[10]

J. Chabrowski, On multiple solutions for the nonhomogeneous p-Laplacian with a critical Sobolev exponent, Differential Integral Equations, 8 (1995), 705-716.

[11]

M. ClappO. Kavian and B. Ruf, Multiple solutions of nonhomogeneous elliptic equations with critical nonlinearity on symmetric domains, Communications in Contemporary Mathematics, 5 (2003), 147-169. doi: 10.1142/S0219199703000963.

[12]

T. K. Donaldson and N. S. Trudinger, Orlicz-Sobolev spaces and imbedding theorems, J. Functional Analysis, 8 (1971), 52-75.

[13]

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

[14]

G. M. Figueiredo and H. Ramos Quoirin, Ground states of elliptic problems involving non-homogeneous operators, Indiana Univ. Math. J., 65 (2016), 779-795. doi: 10.1512/iumj.2016.65.5828.

[15]

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

[16]

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

[17]

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

[18]

J. P. Gossez, Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, in Nonlinear Analysis, Function Spaces and Applications, Proc. Spring School, Horni Bradlo, 1978, Teubner, Leipzig, 59-94, 1979.

[19]

N. Hirano and N. Shioji, A multiplicity result including a sign-changing solution for an inhomogeneous Neumann problem with critical exponent, Proc. Royal Soc. Edinburgh Sect. A, 137 (2007), 333-347. doi: 10.1017/S0308210505001277.

[20]

D. Motreanu and M. Tanaka, Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter, Annali di Matematica Pura ed Applicata, 193 (2014), 1255-1282. doi: 10.1007/s10231-013-0327-9.

[21]

D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear (p, q)-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937. doi: 10.1090/S0002-9947-2013-06124-7.

[22]

M. N. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1985.

[23]

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.

[24]

M. Tanaka, Generalized eigenvalue problems for (p, q)-Laplacian with indefinite weight, J. Math. Anal. Appl., 419 (2014), 1181-1192. doi: 10.1016/j.jmaa.2014.05.044.

[25]

G. Tarantello, On nonhomogenous elliptic equations involving critical Sobolev expoent, I. H. Poincaré, Analyse Non Linéaire, 9 (1992), 281-304. doi: 10.1016/S0294-1449(16)30238-4.

[26]

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

[27]

M. Willem, Minimax Theorems, Birkhäuser Boston, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[28]

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.

[29]

T.-F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\mathbb{R}^N$ involving sign-changing weight, J. Functional Analysis, 258 (2010), 99-131. doi: 10.1016/j.jfa.2009.08.005.

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