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March  2013, 12(2): 815-829. doi: 10.3934/cpaa.2013.12.815

## Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter

 1 Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania 2 Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received  September 2011 Revised  January 2012 Published  September 2012

A nonlinear elliptic equation with $p$-Laplacian, concave-convex reaction term depending on a parameter $\lambda>0$, and homogeneous boundary condition, is investigated. A bifurcation result, which describes the set of positive solutions as $\lambda$ varies, is obtained through variational methods combined with truncation and comparison techniques.
Citation: Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure & Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815
##### References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints,, Mem. Amer. Math. Soc., 196 (2008).   Google Scholar [2] A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems,, J. Funct. Anal., 122 (1994), 519.  doi: 10.1006/jfan.1994.1078.  Google Scholar [3] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplace operator,, Comm. Partial Differential Equations, 31 (2006), 849.  doi: 10.1080/03605300500394447.  Google Scholar [4] D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity,, Bull. Austral. Math. Soc., 77 (2008), 285.  doi: 10.1017/S0004972708000282.  Google Scholar [5] 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 [6] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the $p$-Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 737.  doi: 10.1017/S0308210509000845.  Google Scholar [7] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Ser. Math. Anal. Appl., 9 (2006).   Google Scholar [8] L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,", Ser. Math. Anal. Appl., 8 (2005).   Google Scholar [9] J. P. Garcia Azorero, J. J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, Comm. Contemp. Math., 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [10] M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents,, Nonlinear Anal., 13 (1989), 879.  doi: 10.1016/0362-546X(89)90020-5.  Google Scholar [11] S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin,, Tohoku Math. J., 62 (2010), 137.  doi: 10.2748/tmj/1270041030.  Google Scholar [12] An Lê, Eigenvalue problems for the $p$-Laplacian,, Nonlinear Anal., 64 (2006), 1057.  doi: 10.1016/j.na.2005.05.056.  Google Scholar [13] S. 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 [14] 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 [15] P. Lindqvist, On the equation div$(|\nabla u|^{p-2}\nabla u) +\lambda |u|^{p-2}u=0$, , Proc. Amer. Math. Soc., 109 (1990), 157.  doi: 10.1090/S0002-9939-1990-1007505-7.  Google Scholar [16] O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz condition,, J. Differential Equations, 245 (2008), 3628.  doi: 10.1016/j.jde.2008.02.035.  Google Scholar [17] I. Peral, Some results on quasilinear elliptic equations: growth versus shape,, in, (1998), 153.   Google Scholar [18] 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] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints,, Mem. Amer. Math. Soc., 196 (2008).   Google Scholar [2] A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems,, J. Funct. Anal., 122 (1994), 519.  doi: 10.1006/jfan.1994.1078.  Google Scholar [3] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplace operator,, Comm. Partial Differential Equations, 31 (2006), 849.  doi: 10.1080/03605300500394447.  Google Scholar [4] D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity,, Bull. Austral. Math. Soc., 77 (2008), 285.  doi: 10.1017/S0004972708000282.  Google Scholar [5] 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 [6] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the $p$-Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 737.  doi: 10.1017/S0308210509000845.  Google Scholar [7] L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", Ser. Math. Anal. Appl., 9 (2006).   Google Scholar [8] L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems,", Ser. Math. Anal. Appl., 8 (2005).   Google Scholar [9] J. P. Garcia Azorero, J. J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, Comm. Contemp. Math., 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [10] M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents,, Nonlinear Anal., 13 (1989), 879.  doi: 10.1016/0362-546X(89)90020-5.  Google Scholar [11] S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin,, Tohoku Math. J., 62 (2010), 137.  doi: 10.2748/tmj/1270041030.  Google Scholar [12] An Lê, Eigenvalue problems for the $p$-Laplacian,, Nonlinear Anal., 64 (2006), 1057.  doi: 10.1016/j.na.2005.05.056.  Google Scholar [13] S. 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 [14] 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 [15] P. Lindqvist, On the equation div$(|\nabla u|^{p-2}\nabla u) +\lambda |u|^{p-2}u=0$, , Proc. Amer. Math. Soc., 109 (1990), 157.  doi: 10.1090/S0002-9939-1990-1007505-7.  Google Scholar [16] O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz condition,, J. Differential Equations, 245 (2008), 3628.  doi: 10.1016/j.jde.2008.02.035.  Google Scholar [17] I. Peral, Some results on quasilinear elliptic equations: growth versus shape,, in, (1998), 153.   Google Scholar [18] 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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