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 - A, 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. 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. 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. 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. 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, (1958). Google Scholar

[6]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis,, Series in Mathematical Analysis and Applications, (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. 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. 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. 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. doi: 10.1186/1687-2770-2012-152. Google Scholar

[11]

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

[19]

P. Pucci and J. Serrin, The Maximum Principle,, Progress in Nonlinear Differential Equations and their Applications, (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. 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. 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. 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. 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. 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, (1958). Google Scholar

[6]

L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis,, Series in Mathematical Analysis and Applications, (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. 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. 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. 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. doi: 10.1186/1687-2770-2012-152. Google Scholar

[11]

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

[19]

P. Pucci and J. Serrin, The Maximum Principle,, Progress in Nonlinear Differential Equations and their Applications, (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. doi: 10.1016/j.jmaa.2011.08.030. Google Scholar

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