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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[26]

R. 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 S. Kyritsi, Handbook of Applied Analysis Springer, Dordrecht, 2009. doi: 10.1007/b120946. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[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. Google Scholar

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[26]

R. 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 S. Kyritsi, Handbook of Applied Analysis Springer, Dordrecht, 2009. doi: 10.1007/b120946. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

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