May  2014, 34(5): 2037-2060. doi: 10.3934/dcds.2014.34.2037

Dirichlet $(p,q)$-equations at resonance

1. 

Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków

2. 

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

Received  January 2013 Revised  July 2013 Published  October 2013

We consider a parametric nonlinear Dirichlet equation driven by the sum of a $p$-Laplacian and a $q$-Laplacian ($1 < q < p < +\infty$, $p ≥ 2$) and with a Carathéodory reaction which at $\pm\infty$ is resonant with respect to the principal eigenvalue $\widehat{\lambda}_1(p) > 0$ of $(-\Delta_p, W^{1,p}_0(\Omega))$. Using critical point theory, truncation and comparison techniques and critical groups (Morse theory), we show that for all small values of the parameter $\lambda>0$, the problem has at least five nontrivial solutions, four of constant sign (two positive and two negative) and the fifth nodal (sign-changing).
Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037
References:
[1]

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

[2]

L. Cherfils and V. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with $p& q$ Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22.  Google Scholar

[3]

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

[4]

J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, (French) [Existence and uniqueness of positive solutions of some quasilinear elliptic equations] C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.  Google Scholar

[5]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, With the assistance of William G. Bade and Robert G. Bartle. Reprint of the 1958 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[6]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9. Chapman and Hall/ CRC Press, Boca Raton, FL, 2006.  Google Scholar

[7]

L. Gasiński and N. S. Papageorgiou, Nodal and multiple constant sign solutions for resonant $p$-Laplacian equations with a nonsmooth potential, Nonlinear Anal., 71 (2009), 5747-5772. doi: 10.1016/j.na.2009.04.063.  Google Scholar

[8]

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

[9]

L. Gasiński and N. S. Papageorgiou, Nonhomogeneous nonlinear Dirichlet problems with a $p$-superlinear reaction, Abstr. Appl. Anal., 2012 (2012), 1-28, Article ID 918271. doi: 10.1155/2012/918271.  Google Scholar

[10]

L. Gasiński and N. S. Papageorgiou, Multiplicity of positive solutions for eigenvalue problems of $(p,2)$-equations, Boundary Value Problems, 152 (2012), 1-17. doi: 10.1186/1687-2770-2012-152.  Google Scholar

[11]

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, 2003.  Google Scholar

[12]

Q.-S. Jiu and J.-B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.  Google Scholar

[13]

O. A. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London 1968.  Google Scholar

[14]

G. M. Lieberman, The natural generalizations of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.  Google Scholar

[15]

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

[16]

J.-Q. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.  Google Scholar

[17]

E. Medeiros and K. Perera, Multiplicity of solutions for a quasilinear elliptic problems via the cohomological index, Nonlinear Anal., 71 (2009), 3654-3660. doi: 10.1016/j.na.2009.02.013.  Google Scholar

[18]

V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Dedicated to Olga Ladyzhenskaya, Topol. Methods Nonlinear Anal., 10 (1997), 387-397.  Google Scholar

[19]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[20]

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

show all references

References:
[1]

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

[2]

L. Cherfils and V. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with $p& q$ Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22.  Google Scholar

[3]

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

[4]

J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, (French) [Existence and uniqueness of positive solutions of some quasilinear elliptic equations] C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.  Google Scholar

[5]

N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, With the assistance of William G. Bade and Robert G. Bartle. Reprint of the 1958 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[6]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9. Chapman and Hall/ CRC Press, Boca Raton, FL, 2006.  Google Scholar

[7]

L. Gasiński and N. S. Papageorgiou, Nodal and multiple constant sign solutions for resonant $p$-Laplacian equations with a nonsmooth potential, Nonlinear Anal., 71 (2009), 5747-5772. doi: 10.1016/j.na.2009.04.063.  Google Scholar

[8]

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

[9]

L. Gasiński and N. S. Papageorgiou, Nonhomogeneous nonlinear Dirichlet problems with a $p$-superlinear reaction, Abstr. Appl. Anal., 2012 (2012), 1-28, Article ID 918271. doi: 10.1155/2012/918271.  Google Scholar

[10]

L. Gasiński and N. S. Papageorgiou, Multiplicity of positive solutions for eigenvalue problems of $(p,2)$-equations, Boundary Value Problems, 152 (2012), 1-17. doi: 10.1186/1687-2770-2012-152.  Google Scholar

[11]

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics. Springer, New York, 2003.  Google Scholar

[12]

Q.-S. Jiu and J.-B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.  Google Scholar

[13]

O. A. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London 1968.  Google Scholar

[14]

G. M. Lieberman, The natural generalizations of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. doi: 10.1080/03605309108820761.  Google Scholar

[15]

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

[16]

J.-Q. Liu and J. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.  Google Scholar

[17]

E. Medeiros and K. Perera, Multiplicity of solutions for a quasilinear elliptic problems via the cohomological index, Nonlinear Anal., 71 (2009), 3654-3660. doi: 10.1016/j.na.2009.02.013.  Google Scholar

[18]

V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Dedicated to Olga Ladyzhenskaya, Topol. Methods Nonlinear Anal., 10 (1997), 387-397.  Google Scholar

[19]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[20]

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

[1]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395

[2]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897

[3]

Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921

[4]

Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235

[5]

Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883

[6]

Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Positive and nodal solutions for parametric nonlinear Robin problems with indefinite potential. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6133-6166. doi: 10.3934/dcds.2016068

[7]

Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125

[8]

Shuhong Chen, Zhong Tan. Optimal partial regularity results for nonlinear elliptic systems in Carnot groups. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3391-3405. doi: 10.3934/dcds.2013.33.3391

[9]

M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057

[10]

Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151

[11]

Vladimir Lubyshev. Precise range of the existence of positive solutions of a nonlinear, indefinite in sign Neumann problem. Communications on Pure & Applied Analysis, 2009, 8 (3) : 999-1018. doi: 10.3934/cpaa.2009.8.999

[12]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253

[13]

Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335

[14]

Geng Chen, Yannan Shen. Existence and regularity of solutions in nonlinear wave equations. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3327-3342. doi: 10.3934/dcds.2015.35.3327

[15]

Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155

[16]

Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052

[17]

Kazumasa Fujiwara, Tohru Ozawa. On the lifespan of strong solutions to the periodic derivative nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 275-280. doi: 10.3934/eect.2018013

[18]

Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735

[19]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[20]

Kun Cheng, Yinbin Deng. Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 77-103. doi: 10.3934/dcds.2017004

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]