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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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