April  2018, 11(2): 155-178. doi: 10.3934/dcdss.2018010

Multiple solutions for nonlinear nonhomogeneous resonant coercive problems

1. 

Dipartimento di Matematica e Informatica, Università degli studi di Palermo, Via Archirafi, 90123 -Palermo, Italy

2. 

Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

* Corresponding author: E. Tornatore

Received  December 2016 Revised  April 2017 Published  January 2018

Fund Project: The authors Averna, Papageorgiou and Tornatore were partially supported by INdAM -GNAMPA Project 2016.

We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a $p$-Laplacian ($2<p$) and a Laplacian. The reaction term is a Carathéodory function $f(z,x)$ which is resonant with respect to the principal eigenvalue of ($-\Delta_p,\, W^{1,p}_0(\Omega)$). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of $f(z,\cdot)$ near zero. By strengthening the regularity of $f(z,\cdot)$, we are able to generate a second nodal solution for a total of four nontrivial smooth solutions.

Citation: Diego Averna, Nikolaos S. Papageorgiou, Elisabetta Tornatore. Multiple solutions for nonlinear nonhomogeneous resonant coercive problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 155-178. doi: 10.3934/dcdss.2018010
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality contraints, Memoirs Amer. Math. Soc. , 196 (2008), No. 915, ⅵ+70 pp. doi: 10.1090/memo/0915.

[2]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Nodal solutions for nonlinear nonhomogeneous Neumann equations, Topol. Methods Nonlin. Anal., 43 (2014), 421-438.  doi: 10.12775/TMNA.2014.025.

[3]

S. AizicoviciN.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]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.

[5]

V. BenciP. D'AveniaD. Fortunato and L. Pisani, Solutions in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324.  doi: 10.1007/s002050000101.

[6]

P. CanditoR. Livrea and N. S. Papageorgiou, Nonlinear elliptic equations with asymmetric asymptotic behavior at $± ∞$, Nonlinear Anal. Real World Appl., 32 (2016), 159-177.  doi: 10.1016/j.nonrwa.2016.04.005.

[7]

P. Candito, R. Livrea and N. S. Papageorgiou, Nonlinear nonhomogeneous Neumann eigenvalue problems, Electron. J. Qual. Theory Differ. Equ. , 46 (2015), 24 pp.

[8]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.

[9]

L. Cherfils and Y. Ilyasov, On the stationary solutions of generalized solutions of generalized reaction diffusion equation with $p$, $q$ Laplacian, Comm. Pure Appl. Anal., 4 (2005), 9-22. 

[10]

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.

[11]

S. Cingolani and G. Vannella, Critical groups computations on a class of Sobolev Banach space via Morse index, Ann. Inst. H. Poincare -Analyse Nonlineaire, 20 (2003), 271-292.  doi: 10.1016/S0294-1449(02)00011-2.

[12]

M. 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.

[13]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton FL, 2006.

[14]

L. Gasinski and N. S. Papageorgiou, Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set-Valued Anal., 20 (2012), 417-443.  doi: 10.1007/s11228-011-0198-4.

[15]

L. Gasinski and N. S. Papageorgiou, Nonlinear elliptic equations with a jumping reaction, J. Math. Anal. Appl., 443 (2016), 1033-1070.  doi: 10.1016/j.jmaa.2016.05.053.

[16]

L. Gasinski and N. S. Papageorgiou, Positive solutions for the Robin $p$-Laplacian problem with competing nonlinearities, Adv. Calc. Variations -to appear.

[17]

L. Gasinski and N. S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis Springer, Heidelberg, 2016. doi: 10.1007/978-3-319-27817-9.

[18]

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

[19]

S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear coercive Neumann problems, Comm. Pure Appl. Anal., 8 (2009), 1957-1974.  doi: 10.3934/cpaa.2009.8.1957.

[20]

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations Academic Press, New York, 1968.

[21]

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

[22]

S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236.  doi: 10.1016/j.jmaa.2005.04.034.

[23]

J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600.  doi: 10.1112/S0024609304004023.

[24]

E. Michael, Continuous selections Ⅰ, Annals Math., 63 (1956), 361-382.  doi: 10.2307/1969615.

[25]

D. Motreanu, V. 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. 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 S. Kyritsi, Handbook of Applied Analysis Springer, Dordrecht, 2009. doi: 10.1007/b120946.

[28]

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

[29]

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

[30]

N. S. Papageorgiou and G. Smyrlis, On nonlinear nonhomogeneous Dirichlet equation, Pacific J. Math., 264 (2013), 421-453.  doi: 10.2140/pjm.2013.264.421.

[31]

N. S. Papageorgiou and P. Winkert, Resonant ($p,2$)-equation with concave terms, Applicable Analysis, 94 (2015), 342-360.  doi: 10.1080/00036811.2014.895332.

[32]

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

[33]

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

[34]

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

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 contraints, Memoirs Amer. Math. Soc. , 196 (2008), No. 915, ⅵ+70 pp. doi: 10.1090/memo/0915.

[2]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Nodal solutions for nonlinear nonhomogeneous Neumann equations, Topol. Methods Nonlin. Anal., 43 (2014), 421-438.  doi: 10.12775/TMNA.2014.025.

[3]

S. AizicoviciN.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]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.

[5]

V. BenciP. D'AveniaD. Fortunato and L. Pisani, Solutions in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324.  doi: 10.1007/s002050000101.

[6]

P. CanditoR. Livrea and N. S. Papageorgiou, Nonlinear elliptic equations with asymmetric asymptotic behavior at $± ∞$, Nonlinear Anal. Real World Appl., 32 (2016), 159-177.  doi: 10.1016/j.nonrwa.2016.04.005.

[7]

P. Candito, R. Livrea and N. S. Papageorgiou, Nonlinear nonhomogeneous Neumann eigenvalue problems, Electron. J. Qual. Theory Differ. Equ. , 46 (2015), 24 pp.

[8]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problem Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.

[9]

L. Cherfils and Y. Ilyasov, On the stationary solutions of generalized solutions of generalized reaction diffusion equation with $p$, $q$ Laplacian, Comm. Pure Appl. Anal., 4 (2005), 9-22. 

[10]

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.

[11]

S. Cingolani and G. Vannella, Critical groups computations on a class of Sobolev Banach space via Morse index, Ann. Inst. H. Poincare -Analyse Nonlineaire, 20 (2003), 271-292.  doi: 10.1016/S0294-1449(02)00011-2.

[12]

M. 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.

[13]

L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton FL, 2006.

[14]

L. Gasinski and N. S. Papageorgiou, Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set-Valued Anal., 20 (2012), 417-443.  doi: 10.1007/s11228-011-0198-4.

[15]

L. Gasinski and N. S. Papageorgiou, Nonlinear elliptic equations with a jumping reaction, J. Math. Anal. Appl., 443 (2016), 1033-1070.  doi: 10.1016/j.jmaa.2016.05.053.

[16]

L. Gasinski and N. S. Papageorgiou, Positive solutions for the Robin $p$-Laplacian problem with competing nonlinearities, Adv. Calc. Variations -to appear.

[17]

L. Gasinski and N. S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis Springer, Heidelberg, 2016. doi: 10.1007/978-3-319-27817-9.

[18]

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

[19]

S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear coercive Neumann problems, Comm. Pure Appl. Anal., 8 (2009), 1957-1974.  doi: 10.3934/cpaa.2009.8.1957.

[20]

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations Academic Press, New York, 1968.

[21]

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

[22]

S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236.  doi: 10.1016/j.jmaa.2005.04.034.

[23]

J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600.  doi: 10.1112/S0024609304004023.

[24]

E. Michael, Continuous selections Ⅰ, Annals Math., 63 (1956), 361-382.  doi: 10.2307/1969615.

[25]

D. Motreanu, V. 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. 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 S. Kyritsi, Handbook of Applied Analysis Springer, Dordrecht, 2009. doi: 10.1007/b120946.

[28]

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

[29]

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

[30]

N. S. Papageorgiou and G. Smyrlis, On nonlinear nonhomogeneous Dirichlet equation, Pacific J. Math., 264 (2013), 421-453.  doi: 10.2140/pjm.2013.264.421.

[31]

N. S. Papageorgiou and P. Winkert, Resonant ($p,2$)-equation with concave terms, Applicable Analysis, 94 (2015), 342-360.  doi: 10.1080/00036811.2014.895332.

[32]

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

[33]

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

[34]

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

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