# American Institute of Mathematical Sciences

January  2014, 13(1): 203-215. doi: 10.3934/cpaa.2014.13.203

## A pair of positive solutions for $(p,q)$-equations with combined nonlinearities

 1 Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków 2 Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received  October 2012 Revised  April 2013 Published  July 2013

We consider a parametric nonlinear Dirichlet problem driven by the $(p,q)$-differential operator, with a reaction consisting of a concave'' term perturbed by a $(p-1)$-superlinear perturbation, which need not satisfy the Ambrosetti-Rabinowitz condition (problem with combined or competing nonlinearities). Using variational methods we show that for small values of the parameter the problem has at least two nontrivial positive smooth solutions.
Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. A pair of positive solutions for $(p,q)$-equations with combined nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 203-215. doi: 10.3934/cpaa.2014.13.203
##### References:
 [1] A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, J. Funct. Anal., 122 (1994), 519.  doi: 10.1006/jfan.1994.1078.  Google Scholar [2] V. Benci, D. Fortunato and L. Pisani, Solitons like solutions of a Lorentz invariant equation in dimension $3$,, Rev. Math. Phys., 10 (1998), 315.  doi: 10.1142/S0129055X98000100.  Google Scholar [3] L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents,, Nonlinear Anal., 24 (1995), 1639.  doi: 10.1016/0362-546X(94)E0054-K.  Google Scholar [4] 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 [5] G. M. Figueiredo, Existence of positive solutions for a class of $(p,q)$-elliptic problems with critical growth on $\mathbbR^N$,, J. Math. Anal. Appl., 378 (2011), 507.  doi: 10.1016/j.jmaa.2011.02.017.  Google Scholar [6] M. Filippakis, L. Gasiński and N. S. Papageorgiou, On the existence of positive solutions for hemivariational inequalities driven by the $p$-Laplacian,, J. Global Optim., 31 (2005), 173.  doi: 10.1007/s10898-003-5444-3.  Google Scholar [7] J. García Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, Commun. Contemp. Math., 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [8] L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,", Chapman and Hall/ CRC Press, (2005).   Google Scholar [9] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman and Hall/ CRC Press, (2006).   Google Scholar [10] 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 [11] 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 [12] Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations,, J. Math. Anal. Appl., 286 (2003), 32.  doi: 10.1016/S0022-247X(03)00282-8.  Google Scholar [13] S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin,, T\^ohoku Math. J., 62 (2010), 137.  doi: 10.2748/tmj/1270041030.  Google Scholar [14] O. A. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", Vol. 46 of Mathematics in Science and Engineering, (1968).   Google Scholar [15] 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 [16] S.-J. Li, S. Wu and H.-S. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities,, J. Differential Equations, 185 (2002), 200.  doi: 10.1006/jdeq.2001.4167.  Google Scholar [17] G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of $p$-Laplacian type without the Ambrosetti-Rabinowitz condition,, Nonlinear Anal., 72 (2010), 4602.  doi: 10.1016/j.na.2010.02.037.  Google Scholar [18] G. Li and G. Zhang, Multiple solutions for the p&g-Laplacian problem with critical exponent,, Acta Math. Sci. Ser. B Engl. Ed., 29B (2009), 903.  doi: 10.1016/S0252-9602(09)60089-8.  Google Scholar [19] 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 [20] P. Pucci and J. Serrin, "The Maximum Principle,", Birkh{\, (2007).   Google Scholar [21] 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 [22] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar

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##### References:
 [1] A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, J. Funct. Anal., 122 (1994), 519.  doi: 10.1006/jfan.1994.1078.  Google Scholar [2] V. Benci, D. Fortunato and L. Pisani, Solitons like solutions of a Lorentz invariant equation in dimension $3$,, Rev. Math. Phys., 10 (1998), 315.  doi: 10.1142/S0129055X98000100.  Google Scholar [3] L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents,, Nonlinear Anal., 24 (1995), 1639.  doi: 10.1016/0362-546X(94)E0054-K.  Google Scholar [4] 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 [5] G. M. Figueiredo, Existence of positive solutions for a class of $(p,q)$-elliptic problems with critical growth on $\mathbbR^N$,, J. Math. Anal. Appl., 378 (2011), 507.  doi: 10.1016/j.jmaa.2011.02.017.  Google Scholar [6] M. Filippakis, L. Gasiński and N. S. Papageorgiou, On the existence of positive solutions for hemivariational inequalities driven by the $p$-Laplacian,, J. Global Optim., 31 (2005), 173.  doi: 10.1007/s10898-003-5444-3.  Google Scholar [7] J. García Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, Commun. Contemp. Math., 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [8] L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,", Chapman and Hall/ CRC Press, (2005).   Google Scholar [9] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman and Hall/ CRC Press, (2006).   Google Scholar [10] 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 [11] 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 [12] Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations,, J. Math. Anal. Appl., 286 (2003), 32.  doi: 10.1016/S0022-247X(03)00282-8.  Google Scholar [13] S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin,, T\^ohoku Math. J., 62 (2010), 137.  doi: 10.2748/tmj/1270041030.  Google Scholar [14] O. A. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", Vol. 46 of Mathematics in Science and Engineering, (1968).   Google Scholar [15] 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 [16] S.-J. Li, S. Wu and H.-S. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities,, J. Differential Equations, 185 (2002), 200.  doi: 10.1006/jdeq.2001.4167.  Google Scholar [17] G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of $p$-Laplacian type without the Ambrosetti-Rabinowitz condition,, Nonlinear Anal., 72 (2010), 4602.  doi: 10.1016/j.na.2010.02.037.  Google Scholar [18] G. Li and G. Zhang, Multiple solutions for the p&g-Laplacian problem with critical exponent,, Acta Math. Sci. Ser. B Engl. Ed., 29B (2009), 903.  doi: 10.1016/S0252-9602(09)60089-8.  Google Scholar [19] 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 [20] P. Pucci and J. Serrin, "The Maximum Principle,", Birkh{\, (2007).   Google Scholar [21] 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 [22] J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations,, Appl. Math. Optim., 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar
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