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September  2016, 15(5): 1545-1570. doi: 10.3934/cpaa.2016002

## Positive solutions for parametric $p$-Laplacian equations

 1 Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780 2 Technological Educational Institute of Athens, Department of Mathematics, Athens 12210, Greece

Received  January 2014 Revised  June 2016 Published  July 2016

We consider parametric equations driven by the $p$ - Laplacian and with a reaction which has a $p$ - logistic form or is the sum of two competing nonlinearities (concave-convex nonlinearities). We look for positive solutions and how their solution set depends on the parameter $\lambda>0$. For the $p$ - logistic equation, we examine the subdiffusive, equidiffusive and superdiffusive cases. For the equations with competing nonlinearities, we consider the case of the sum of a concave'' (i.e., $(p-1)$ - sublinear) term and of a convex'' (i.e., $(p-1)$ - superlinear) term. For the latter, we do not assume the usual in such cases Ambrosetti-Rabinowitz condition. Our approach is variational based on the critical point theory combined with suitable truncation and comparison techniques.
Citation: Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002
##### References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiple positive solutions for a $p$-Laplacian Dirichlet problem with a superdiffusive reaction,, \emph{Houston J. Math.}, 36 (2010), 313.   Google Scholar [2] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplacian operator,, \emph{Comm. Partial Differential Equations}, 31 (2006), 849.  doi: 10.1080/03605300500394447.  Google Scholar [3] G.Barletta, R.Livrea and N. S. Papageorgiou, A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian,, Comm. Pure Appl. Anal., 13 (2014), 1075.   Google Scholar [4] G.D'Agui, S.Marano and N. S. Papageorgiou, Multiple solutions to a Neumann problem with equidiffusive reaction,, Disc. Cont. Dyn. Syst-Ser. S\textbf{5} (2012), 5 (2012), 765.   Google Scholar [5] Y. Dong, A priori estimates and existence of positive solutions for a quasilinear elliptic equation,, \emph{J. London Math. Soc.}, 72 (2005), 645.  doi: 10.1112/S0024610705006848.  Google Scholar [6] W. Dong and J. J. Chen, Existence and multiplicity results for a degenerate elliptic equation,, \emph{Acta Math. Sinica (English Series)}, 22 (2006), 665.  doi: 10.1007/s10114-005-0696-0.  Google Scholar [7] J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, \emph{Comm. Contemp. Math.}, 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [8] J. Garcia Melian and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large,, \emph{Differential Intergal Equations}, 13 (2000), 1201.   Google Scholar [9] L. Gasinski and N. S.Papageorgiou, Nonlinear Analysis,, Chapman Hall/CRC, (2006).   Google Scholar [10] L. Gasinski and N. S . Papageorgiou, Bifurcation type results for nonlinear parametric elliptic equations,, \emph{Proc. Royal Soc. Edinburgh}, 142A (2012), 595.  doi: 10.1017/S0308210511000126.  Google Scholar [11] L. Gasinski and N. S. Papageorgiou, Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 1985.  doi: 10.3934/cpaa.2013.12.1985.  Google Scholar [12] L.Gasinski and N. S. Papageorgiou, A pair of positive solutions for $(p,q)$-equations with combined nonlinearities,, Comm. Pure Appl. Anal., 13 (2014), 203.   Google Scholar [13] L.Gasinski and N. S. Papageorgiou, Multiple solutions for a class of nonlinear Neumann eigenvalue problems,, Comm. Pure Appl. Anal., 13 (2014), 1491.   Google Scholar [14] M. Guedda and L. Veron, Bifurcation phenomena associated to the $p$-Laplace operator,, \emph{Trans. Amer. Math. Soc.}, 310 (1988), 419.  doi: 10.2307/2001132.  Google Scholar [15] M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents,, \emph{Nonlinear Anal.}, 13 (1989), 879.  doi: 10.1016/0362-546X(89)90020-5.  Google Scholar [16] Z. Guo and Z. Zhang, $W^{1,p}\;$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations,, \emph{J. Math. Anal. Appl.}, 286 (2003), 32.  doi: 10.1016/S0022-247X(03)00282-8.  Google Scholar [17] M. E. Gurtin and R. C. Mac Camy, On the diffusion of biological populations,, \emph{Math. Biosci.}, 33 (1977), 35.   Google Scholar [18] S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$ -Laplacian equations with nonlinearity concave near the origin,, \emph{Tohoku Math. J.}, 62 (2010), 137.  doi: 10.2748/tmj/1270041030.  Google Scholar [19] S. Hu and N. S. Papageorgiou, Double resonance for Dirichlet problems with unbounded and indefinite potential and competing nonlinearities,, \emph{Comm. Pure Appl. Anal.}, 11 (2012), 2005.  doi: 10.3934/cpaa.2012.11.2005.  Google Scholar [20] S. Hu and N.S.Papageorgiou, Nonlinear Neumann problems with indefinite potential and concave terms,, Comm. Pure Appl. Anal., 14 (2015), 2561.   Google Scholar [21] A. Iannizzotto and N. S. Papageorgiou, Positive solutions for generalized nonlinear logistic equations of superdiffusive type,, \emph{Topol. Meth. Nonlin. Anal.}, 38 (2011), 95.   Google Scholar [22] S. Kamin and L. Veron, Flat core properties associated to the $p$-Laplace operator,, \emph{Proc. Amer. Math. Soc.}, 118 (1993), 1079.  doi: 10.2307/2160060.  Google Scholar [23] S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term,, \emph{Discr. Cont. Dynam. Systems}, 33 (2013), 2469.   Google Scholar [24] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968).   Google Scholar [25] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 815.  doi: 10.3934/cpaa.2013.12.815.  Google Scholar [26] G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear boundary value problem of $p$- Laplacian type without the Ambrosetti-Rabinowitz condition,, \emph{Nonlinear Anal.}, 72 (2010), 4602.  doi: 10.1016/j.na.2010.02.037.  Google Scholar [27] N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis,, Springer, (2009).  doi: 10.1007/b120946.  Google Scholar [28] N. S. Papageorgiou and G. Smyrlis, Nonlinear elliptic equations with asymptotically linear reaction term,, \emph{Nonlinear Anal.}, 71 (2009), 3129.  doi: 10.1016/j.na.2009.01.224.  Google Scholar [29] N. S. Papageorgiou and G. Smyrlis, Positive solutions for nonlinear Neumann problems with concave and convex terms,, \emph{Positivity}, 16 (2012), 271.  doi: 10.1007/s11117-011-0124-x.  Google Scholar [30] N. S. Papageorgiou and V. Radulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Neumann and Robin problems with competing nonlinearities,, Dist. Cont. Dyn. Syst., 35 (2015), 5003.   Google Scholar [31] V. Radulescu and D. Repovs, Combined effects in noninear problems arising in the study of anisotropic continuous media,, \emph{Nonlinear Anal.}, 75 (2012), 1524.  doi: 10.1016/j.na.2011.01.037.  Google Scholar [32] S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction,, \emph{Proc. Amer. Math. Soc.}, 129 (2001), 433.  doi: 10.1090/S0002-9939-00-05723-3.  Google Scholar [33] S. Takeuchi, Multiplicity result for a degenerate elliptic equation with logistic reaction,, \emph{J. Differential Equations}, 173 (2001), 138.  doi: 10.1006/jdeq.2000.3914.  Google Scholar [34] S. Takeuchi and Y. Yamada, Asymptotic properties of a reaction-diffusion equation with a degenerate $p$-Laplacian,, \emph{Nonlinear Anal.}, 42 (2000), 41.  doi: 10.1016/S0362-546X(98)00329-0.  Google Scholar [35] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations,, \emph{Appl. Math. Optim.}, 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar

show all references

##### References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiple positive solutions for a $p$-Laplacian Dirichlet problem with a superdiffusive reaction,, \emph{Houston J. Math.}, 36 (2010), 313.   Google Scholar [2] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplacian operator,, \emph{Comm. Partial Differential Equations}, 31 (2006), 849.  doi: 10.1080/03605300500394447.  Google Scholar [3] G.Barletta, R.Livrea and N. S. Papageorgiou, A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian,, Comm. Pure Appl. Anal., 13 (2014), 1075.   Google Scholar [4] G.D'Agui, S.Marano and N. S. Papageorgiou, Multiple solutions to a Neumann problem with equidiffusive reaction,, Disc. Cont. Dyn. Syst-Ser. S\textbf{5} (2012), 5 (2012), 765.   Google Scholar [5] Y. Dong, A priori estimates and existence of positive solutions for a quasilinear elliptic equation,, \emph{J. London Math. Soc.}, 72 (2005), 645.  doi: 10.1112/S0024610705006848.  Google Scholar [6] W. Dong and J. J. Chen, Existence and multiplicity results for a degenerate elliptic equation,, \emph{Acta Math. Sinica (English Series)}, 22 (2006), 665.  doi: 10.1007/s10114-005-0696-0.  Google Scholar [7] J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,, \emph{Comm. Contemp. Math.}, 2 (2000), 385.  doi: 10.1142/S0219199700000190.  Google Scholar [8] J. Garcia Melian and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large,, \emph{Differential Intergal Equations}, 13 (2000), 1201.   Google Scholar [9] L. Gasinski and N. S.Papageorgiou, Nonlinear Analysis,, Chapman Hall/CRC, (2006).   Google Scholar [10] L. Gasinski and N. S . Papageorgiou, Bifurcation type results for nonlinear parametric elliptic equations,, \emph{Proc. Royal Soc. Edinburgh}, 142A (2012), 595.  doi: 10.1017/S0308210511000126.  Google Scholar [11] L. Gasinski and N. S. Papageorgiou, Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 1985.  doi: 10.3934/cpaa.2013.12.1985.  Google Scholar [12] L.Gasinski and N. S. Papageorgiou, A pair of positive solutions for $(p,q)$-equations with combined nonlinearities,, Comm. Pure Appl. Anal., 13 (2014), 203.   Google Scholar [13] L.Gasinski and N. S. Papageorgiou, Multiple solutions for a class of nonlinear Neumann eigenvalue problems,, Comm. Pure Appl. Anal., 13 (2014), 1491.   Google Scholar [14] M. Guedda and L. Veron, Bifurcation phenomena associated to the $p$-Laplace operator,, \emph{Trans. Amer. Math. Soc.}, 310 (1988), 419.  doi: 10.2307/2001132.  Google Scholar [15] M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents,, \emph{Nonlinear Anal.}, 13 (1989), 879.  doi: 10.1016/0362-546X(89)90020-5.  Google Scholar [16] Z. Guo and Z. Zhang, $W^{1,p}\;$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations,, \emph{J. Math. Anal. Appl.}, 286 (2003), 32.  doi: 10.1016/S0022-247X(03)00282-8.  Google Scholar [17] M. E. Gurtin and R. C. Mac Camy, On the diffusion of biological populations,, \emph{Math. Biosci.}, 33 (1977), 35.   Google Scholar [18] S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$ -Laplacian equations with nonlinearity concave near the origin,, \emph{Tohoku Math. J.}, 62 (2010), 137.  doi: 10.2748/tmj/1270041030.  Google Scholar [19] S. Hu and N. S. Papageorgiou, Double resonance for Dirichlet problems with unbounded and indefinite potential and competing nonlinearities,, \emph{Comm. Pure Appl. Anal.}, 11 (2012), 2005.  doi: 10.3934/cpaa.2012.11.2005.  Google Scholar [20] S. Hu and N.S.Papageorgiou, Nonlinear Neumann problems with indefinite potential and concave terms,, Comm. Pure Appl. Anal., 14 (2015), 2561.   Google Scholar [21] A. Iannizzotto and N. S. Papageorgiou, Positive solutions for generalized nonlinear logistic equations of superdiffusive type,, \emph{Topol. Meth. Nonlin. Anal.}, 38 (2011), 95.   Google Scholar [22] S. Kamin and L. Veron, Flat core properties associated to the $p$-Laplace operator,, \emph{Proc. Amer. Math. Soc.}, 118 (1993), 1079.  doi: 10.2307/2160060.  Google Scholar [23] S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term,, \emph{Discr. Cont. Dynam. Systems}, 33 (2013), 2469.   Google Scholar [24] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968).   Google Scholar [25] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 815.  doi: 10.3934/cpaa.2013.12.815.  Google Scholar [26] G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear boundary value problem of $p$- Laplacian type without the Ambrosetti-Rabinowitz condition,, \emph{Nonlinear Anal.}, 72 (2010), 4602.  doi: 10.1016/j.na.2010.02.037.  Google Scholar [27] N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis,, Springer, (2009).  doi: 10.1007/b120946.  Google Scholar [28] N. S. Papageorgiou and G. Smyrlis, Nonlinear elliptic equations with asymptotically linear reaction term,, \emph{Nonlinear Anal.}, 71 (2009), 3129.  doi: 10.1016/j.na.2009.01.224.  Google Scholar [29] N. S. Papageorgiou and G. Smyrlis, Positive solutions for nonlinear Neumann problems with concave and convex terms,, \emph{Positivity}, 16 (2012), 271.  doi: 10.1007/s11117-011-0124-x.  Google Scholar [30] N. S. Papageorgiou and V. Radulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Neumann and Robin problems with competing nonlinearities,, Dist. Cont. Dyn. Syst., 35 (2015), 5003.   Google Scholar [31] V. Radulescu and D. Repovs, Combined effects in noninear problems arising in the study of anisotropic continuous media,, \emph{Nonlinear Anal.}, 75 (2012), 1524.  doi: 10.1016/j.na.2011.01.037.  Google Scholar [32] S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction,, \emph{Proc. Amer. Math. Soc.}, 129 (2001), 433.  doi: 10.1090/S0002-9939-00-05723-3.  Google Scholar [33] S. Takeuchi, Multiplicity result for a degenerate elliptic equation with logistic reaction,, \emph{J. Differential Equations}, 173 (2001), 138.  doi: 10.1006/jdeq.2000.3914.  Google Scholar [34] S. Takeuchi and Y. Yamada, Asymptotic properties of a reaction-diffusion equation with a degenerate $p$-Laplacian,, \emph{Nonlinear Anal.}, 42 (2000), 41.  doi: 10.1016/S0362-546X(98)00329-0.  Google Scholar [35] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations,, \emph{Appl. Math. Optim.}, 12 (1984), 191.  doi: 10.1007/BF01449041.  Google Scholar
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