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

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  • 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).
    Mathematics Subject Classification: 35J20, 35J60, 35J92, 58E05.


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  • [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.


    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.


    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.


    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.


    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.


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


    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.


    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.


    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.


    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.


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


    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.


    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.


    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.


    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.


    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.


    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.


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


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


    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.

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