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April  2019, 12(2): 347-374. doi: 10.3934/dcdss.2019024

Multiple solutions for (p, 2)-equations at resonance

Nikolaos S. Papageorgiou 1,, , Calogero Vetro 2, and  Francesca Vetro 3,

1. 

Department of Mathematics, National Technical University, Zagrafou Campus, Athens, 15780, Greece
2. 

Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
3. 

Department of Energy, Information Engineering and Mathematical Models, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy

* Corresponding author: Nikolaos Papageorgiou

Dedicated to Professor Vicentiu D. Radulescu with friendship and admiration.

Received  April 2017 Revised  December 2017 Published  August 2018

We consider a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a $p$-Laplacian and a Laplacian and a reaction term which is $(p-1)$-linear near $\pm \infty$ and resonant with respect to any nonprincipal variational eigenvalue of $(-\Delta_p,W^{1,p}_0(\Omega))$. Using variational tools together with truncation and comparison techniques and Morse Theory (critical groups), we establish the existence of six nontrivial smooth solutions. For five of them we provide sign information and order them.

Keywords: Resonance, truncation and comparison techniques, nonlinear regularity, nonlinear maximum principle, constant sign and nodal solutions, critical groups.
Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 58E05.
Citation: Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro. Multiple solutions for (p, 2)-equations at resonance. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 347-374. doi: 10.3934/dcdss.2019024
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196 (2008), ⅵ+70 pp. doi: 10.1090/memo/0915.

[2]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, On p-superlinear equations with a nonhomogeneous differential operator, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 151-175.  doi: 10.1007/s00030-012-0187-9.

[3]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nodal solutions for (p, 2)-equations, Trans. Amer. Math. Soc., 367 (2015), 7343-7372.  doi: 10.1090/S0002-9947-2014-06324-1.

[4]

R. Aris, Mathematical Modelling Techniques, Research Notes in Mathematics, 24. Pitman Boston, 1979.

[5]

V. Benci, P. D'Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Ration. Mech. Anal., 154 (2000), 297-324.  doi: 10.1007/s002050000101.

[6]

K.-C. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, 6. Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0385-8.

[7]

K.-C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.

[8]

L. Cherfils and Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with p & q-Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22. 

[9]

S. Cingolani and M. Degiovanni, Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations, 30 (2005), 1191-1203.  doi: 10.1080/03605300500257594.

[10]

J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524. 

[11]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin, 1979.

[12]

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.

[13]

M. E. Filippakis and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1922.  doi: 10.1016/j.jde.2008.07.004.

[14]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Ser. Math. Anal. Appl., 9 Chapman and Hall/CRC Press, Boca Raton, 2006.

[15]

L. Gasiński and N. S. Papageorgiou, Multiplicity of positive solutions for eigenvalue problems of (p, 2)-equations, Bound. Value Probl., 2012 (2012), 17 pp. doi: 10.1186/1687-2770-2012-152.

[16]

L. Gasiński and N. S. Papageorgiou, Asymmetric (p, 2)-equations with double resonance, Calc. Var., 56 (2017), Art. 88, 23 pp. doi: 10.1007/s00526-017-1180-2.

[17]

L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2. Nonlinear Analysis, Problem Books in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-27817-9.

[18]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[19]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

[20]

Z. Liang, X. Han and A. Li, Some properties and applications related to the (2, p)-Laplacian operator, Bound. Value Probl., 2016 (2016), 17 pp. doi: 10.1186/s13661-016-0567-x.

[21]

Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158.  doi: 10.1016/j.jmaa.2008.12.053.

[22]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[23]

S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with p-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829.  doi: 10.3934/cpaa.2013.12.815.

[24]

M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations, Nonlinear Anal., 37 (1999), 431-448.  doi: 10.1016/S0362-546X(98)00057-1.

[25]

D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[26]

R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16.  doi: 10.1016/0040-9383(66)90002-4.

[27]

N. S. Papageorgiou and V. D. Rǎdulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim., 69 (2014), 393-430.  doi: 10.1007/s00245-013-9227-z.

[28]

N. S. Papageorgiou and V. D. Rǎdulescu, Resonant (p, 2)-equations with asymmetric reaction, Anal. Appl., 13 (2015), 481-506.  doi: 10.1142/S0219530514500134.

[29]

N. S. Papageorgiou and V. D. Rǎdulescu, Noncoercive resonant (p, 2)-equations, Appl. Math. Optim., 76 (2017), 621-639.  doi: 10.1007/s00245-016-9363-3.

[30]

N. S. Papageorgiou and V. D. Rǎdulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.  doi: 10.1016/j.jde.2014.01.010.

[31]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear. Stud., 16 (2016), 737-764.  doi: 10.1515/ans-2016-0023.

[32]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, On a class of parametric (p, 2)-equations, Appl. Math. Optim., 75 (2017), 193-228.  doi: 10.1007/s00245-016-9330-z.

[33]

R. Pei and J. Zhang, Nontrivial solution for asymmetric (p, 2)-Laplacian Dirichlet problem, Bound. Value Probl., 2014 (2014), 15 pp. doi: 10.1186/s13661-014-0241-0.

[34]

P. Pucci and J. Serrin, The Mximum Principle, Birkhäuser Verlag, Basel, 2007.

[35]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.  doi: 10.1016/S0362-546X(00)00221-2.

[36]

M. Sun, Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance, J. Math. Anal. Appl., 386 (2012), 661-668.  doi: 10.1016/j.jmaa.2011.08.030.

[37]

M. Sun, M. Zhang and J. Su, Critical groups at zero and multiple solutions for a quasilinear elliptic equation, J. Math. Anal. Appl., 428 (2015), 696-712.  doi: 10.1016/j.jmaa.2015.03.033.

[38]

D. Yang and C. Bai, Nonlinear elliptic problem of 2-q-Laplacian type with asymmetric nonlinearities, Electron. J. Differential Equations, 2014 (2014), 13 pp.

show all references

Dedicated to Professor Vicentiu D. Radulescu with friendship and admiration.

References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196 (2008), ⅵ+70 pp. doi: 10.1090/memo/0915.

[2]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, On p-superlinear equations with a nonhomogeneous differential operator, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 151-175.  doi: 10.1007/s00030-012-0187-9.

[3]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Nodal solutions for (p, 2)-equations, Trans. Amer. Math. Soc., 367 (2015), 7343-7372.  doi: 10.1090/S0002-9947-2014-06324-1.

[4]

R. Aris, Mathematical Modelling Techniques, Research Notes in Mathematics, 24. Pitman Boston, 1979.

[5]

V. Benci, P. D'Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Ration. Mech. Anal., 154 (2000), 297-324.  doi: 10.1007/s002050000101.

[6]

K.-C. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, 6. Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0385-8.

[7]

K.-C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.

[8]

L. Cherfils and Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with p & q-Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22. 

[9]

S. Cingolani and M. Degiovanni, Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations, 30 (2005), 1191-1203.  doi: 10.1080/03605300500257594.

[10]

J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524. 

[11]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin, 1979.

[12]

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.

[13]

M. E. Filippakis and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations, 245 (2008), 1883-1922.  doi: 10.1016/j.jde.2008.07.004.

[14]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Ser. Math. Anal. Appl., 9 Chapman and Hall/CRC Press, Boca Raton, 2006.

[15]

L. Gasiński and N. S. Papageorgiou, Multiplicity of positive solutions for eigenvalue problems of (p, 2)-equations, Bound. Value Probl., 2012 (2012), 17 pp. doi: 10.1186/1687-2770-2012-152.

[16]

L. Gasiński and N. S. Papageorgiou, Asymmetric (p, 2)-equations with double resonance, Calc. Var., 56 (2017), Art. 88, 23 pp. doi: 10.1007/s00526-017-1180-2.

[17]

L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2. Nonlinear Analysis, Problem Books in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-27817-9.

[18]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[19]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

[20]

Z. Liang, X. Han and A. Li, Some properties and applications related to the (2, p)-Laplacian operator, Bound. Value Probl., 2016 (2016), 17 pp. doi: 10.1186/s13661-016-0567-x.

[21]

Z. Liang and J. Su, Multiple solutions for semilinear elliptic boundary value problems with double resonance, J. Math. Anal. Appl., 354 (2009), 147-158.  doi: 10.1016/j.jmaa.2008.12.053.

[22]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[23]

S. A. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with p-Laplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829.  doi: 10.3934/cpaa.2013.12.815.

[24]

M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations, Nonlinear Anal., 37 (1999), 431-448.  doi: 10.1016/S0362-546X(98)00057-1.

[25]

D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.

[26]

R. S. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1-16.  doi: 10.1016/0040-9383(66)90002-4.

[27]

N. S. Papageorgiou and V. D. Rǎdulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim., 69 (2014), 393-430.  doi: 10.1007/s00245-013-9227-z.

[28]

N. S. Papageorgiou and V. D. Rǎdulescu, Resonant (p, 2)-equations with asymmetric reaction, Anal. Appl., 13 (2015), 481-506.  doi: 10.1142/S0219530514500134.

[29]

N. S. Papageorgiou and V. D. Rǎdulescu, Noncoercive resonant (p, 2)-equations, Appl. Math. Optim., 76 (2017), 621-639.  doi: 10.1007/s00245-016-9363-3.

[30]

N. S. Papageorgiou and V. D. Rǎdulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256 (2014), 2449-2479.  doi: 10.1016/j.jde.2014.01.010.

[31]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear. Stud., 16 (2016), 737-764.  doi: 10.1515/ans-2016-0023.

[32]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, On a class of parametric (p, 2)-equations, Appl. Math. Optim., 75 (2017), 193-228.  doi: 10.1007/s00245-016-9330-z.

[33]

R. Pei and J. Zhang, Nontrivial solution for asymmetric (p, 2)-Laplacian Dirichlet problem, Bound. Value Probl., 2014 (2014), 15 pp. doi: 10.1186/s13661-014-0241-0.

[34]

P. Pucci and J. Serrin, The Mximum Principle, Birkhäuser Verlag, Basel, 2007.

[35]

J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal., 48 (2002), 881-895.  doi: 10.1016/S0362-546X(00)00221-2.

[36]

M. Sun, Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance, J. Math. Anal. Appl., 386 (2012), 661-668.  doi: 10.1016/j.jmaa.2011.08.030.

[37]

M. Sun, M. Zhang and J. Su, Critical groups at zero and multiple solutions for a quasilinear elliptic equation, J. Math. Anal. Appl., 428 (2015), 696-712.  doi: 10.1016/j.jmaa.2015.03.033.

[38]

D. Yang and C. Bai, Nonlinear elliptic problem of 2-q-Laplacian type with asymmetric nonlinearities, Electron. J. Differential Equations, 2014 (2014), 13 pp.

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