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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 and 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, Houston J. Math., 36 (2010), 313-332. [2] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447. [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), 1075-1086. [4] G.D'Agui, S.Marano and N. S. Papageorgiou, Multiple solutions to a Neumann problem with equidiffusive reaction, Disc. Cont. Dyn. Syst-Ser. S5 (2012), 765-777. [5] Y. Dong, A priori estimates and existence of positive solutions for a quasilinear elliptic equation, J. London Math. Soc., 72 (2005), 645-662. doi: 10.1112/S0024610705006848. [6] W. Dong and J. J. Chen, Existence and multiplicity results for a degenerate elliptic equation, Acta Math. Sinica (English Series), 22 (2006), 665-670. doi: 10.1007/s10114-005-0696-0. [7] J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. doi: 10.1142/S0219199700000190. [8] J. Garcia Melian and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large, Differential Intergal Equations, 13 (2000), 1201-1232. [9] L. Gasinski and N. S.Papageorgiou, Nonlinear Analysis, Chapman Hall/CRC, Boca Raton, 2006. [10] L. Gasinski and N. S . Papageorgiou, Bifurcation type results for nonlinear parametric elliptic equations, Proc. Royal Soc. Edinburgh, 142A (2012), 595-623. doi: 10.1017/S0308210511000126. [11] L. Gasinski and N. S. Papageorgiou, Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential, Comm. Pure Appl. Anal., 12 (2013), 1985-1999. doi: 10.3934/cpaa.2013.12.1985. [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), 203-215. [13] L.Gasinski and N. S. Papageorgiou, Multiple solutions for a class of nonlinear Neumann eigenvalue problems, Comm. Pure Appl. Anal., 13 (2014), 1491-1512. [14] M. Guedda and L. Veron, Bifurcation phenomena associated to the $p$-Laplace operator, Trans. Amer. Math. Soc., 310 (1988), 419-431. doi: 10.2307/2001132. [15] M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902. doi: 10.1016/0362-546X(89)90020-5. [16] 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), 32-50. doi: 10.1016/S0022-247X(03)00282-8. [17] M. E. Gurtin and R. C. Mac Camy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. [18] 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), 137-162. doi: 10.2748/tmj/1270041030. [19] S. Hu and N. S. Papageorgiou, Double resonance for Dirichlet problems with unbounded and indefinite potential and competing nonlinearities, Comm. Pure Appl. Anal., 11 (2012), 2005-2021. doi: 10.3934/cpaa.2012.11.2005. [20] S. Hu and N.S.Papageorgiou, Nonlinear Neumann problems with indefinite potential and concave terms, Comm. Pure Appl. Anal., 14 (2015), 2561-2616. [21] A. Iannizzotto and N. S. Papageorgiou, Positive solutions for generalized nonlinear logistic equations of superdiffusive type, Topol. Meth. Nonlin. Anal., 38 (2011), 95-113. [22] S. Kamin and L. Veron, Flat core properties associated to the $p$-Laplace operator, Proc. Amer. Math. Soc., 118 (1993), 1079-1085. doi: 10.2307/2160060. [23] S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term, Discr. Cont. Dynam. Systems, 33 (2013), 2469-2494. [24] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [25] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829. doi: 10.3934/cpaa.2013.12.815. [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, Nonlinear Anal., 72 (2010), 4602-4613. doi: 10.1016/j.na.2010.02.037. [27] N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis, Springer, New York, 2009. doi: 10.1007/b120946. [28] N. S. Papageorgiou and G. Smyrlis, Nonlinear elliptic equations with asymptotically linear reaction term, Nonlinear Anal., 71 (2009), 3129-3151. doi: 10.1016/j.na.2009.01.224. [29] N. S. Papageorgiou and G. Smyrlis, Positive solutions for nonlinear Neumann problems with concave and convex terms, Positivity, 16 (2012), 271-296. doi: 10.1007/s11117-011-0124-x. [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., A 35 (2015), 5003-5036. [31] V. Radulescu and D. Repovs, Combined effects in noninear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530. doi: 10.1016/j.na.2011.01.037. [32] S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction, Proc. Amer. Math. Soc., 129 (2001), 433-441. doi: 10.1090/S0002-9939-00-05723-3. [33] S. Takeuchi, Multiplicity result for a degenerate elliptic equation with logistic reaction, J. Differential Equations, 173 (2001), 138-144. doi: 10.1006/jdeq.2000.3914. [34] S. Takeuchi and Y. Yamada, Asymptotic properties of a reaction-diffusion equation with a degenerate $p$-Laplacian, Nonlinear Anal., 42 (2000), 41-61. doi: 10.1016/S0362-546X(98)00329-0. [35] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.

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References:
 [1] S. Aizicovici, N. S. Papageorgiou and V. Staicu, Multiple positive solutions for a $p$-Laplacian Dirichlet problem with a superdiffusive reaction, Houston J. Math., 36 (2010), 313-332. [2] D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447. [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), 1075-1086. [4] G.D'Agui, S.Marano and N. S. Papageorgiou, Multiple solutions to a Neumann problem with equidiffusive reaction, Disc. Cont. Dyn. Syst-Ser. S5 (2012), 765-777. [5] Y. Dong, A priori estimates and existence of positive solutions for a quasilinear elliptic equation, J. London Math. Soc., 72 (2005), 645-662. doi: 10.1112/S0024610705006848. [6] W. Dong and J. J. Chen, Existence and multiplicity results for a degenerate elliptic equation, Acta Math. Sinica (English Series), 22 (2006), 665-670. doi: 10.1007/s10114-005-0696-0. [7] J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. doi: 10.1142/S0219199700000190. [8] J. Garcia Melian and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large, Differential Intergal Equations, 13 (2000), 1201-1232. [9] L. Gasinski and N. S.Papageorgiou, Nonlinear Analysis, Chapman Hall/CRC, Boca Raton, 2006. [10] L. Gasinski and N. S . Papageorgiou, Bifurcation type results for nonlinear parametric elliptic equations, Proc. Royal Soc. Edinburgh, 142A (2012), 595-623. doi: 10.1017/S0308210511000126. [11] L. Gasinski and N. S. Papageorgiou, Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential, Comm. Pure Appl. Anal., 12 (2013), 1985-1999. doi: 10.3934/cpaa.2013.12.1985. [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), 203-215. [13] L.Gasinski and N. S. Papageorgiou, Multiple solutions for a class of nonlinear Neumann eigenvalue problems, Comm. Pure Appl. Anal., 13 (2014), 1491-1512. [14] M. Guedda and L. Veron, Bifurcation phenomena associated to the $p$-Laplace operator, Trans. Amer. Math. Soc., 310 (1988), 419-431. doi: 10.2307/2001132. [15] M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902. doi: 10.1016/0362-546X(89)90020-5. [16] 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), 32-50. doi: 10.1016/S0022-247X(03)00282-8. [17] M. E. Gurtin and R. C. Mac Camy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. [18] 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), 137-162. doi: 10.2748/tmj/1270041030. [19] S. Hu and N. S. Papageorgiou, Double resonance for Dirichlet problems with unbounded and indefinite potential and competing nonlinearities, Comm. Pure Appl. Anal., 11 (2012), 2005-2021. doi: 10.3934/cpaa.2012.11.2005. [20] S. Hu and N.S.Papageorgiou, Nonlinear Neumann problems with indefinite potential and concave terms, Comm. Pure Appl. Anal., 14 (2015), 2561-2616. [21] A. Iannizzotto and N. S. Papageorgiou, Positive solutions for generalized nonlinear logistic equations of superdiffusive type, Topol. Meth. Nonlin. Anal., 38 (2011), 95-113. [22] S. Kamin and L. Veron, Flat core properties associated to the $p$-Laplace operator, Proc. Amer. Math. Soc., 118 (1993), 1079-1085. doi: 10.2307/2160060. [23] S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term, Discr. Cont. Dynam. Systems, 33 (2013), 2469-2494. [24] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [25] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829. doi: 10.3934/cpaa.2013.12.815. [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, Nonlinear Anal., 72 (2010), 4602-4613. doi: 10.1016/j.na.2010.02.037. [27] N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis, Springer, New York, 2009. doi: 10.1007/b120946. [28] N. S. Papageorgiou and G. Smyrlis, Nonlinear elliptic equations with asymptotically linear reaction term, Nonlinear Anal., 71 (2009), 3129-3151. doi: 10.1016/j.na.2009.01.224. [29] N. S. Papageorgiou and G. Smyrlis, Positive solutions for nonlinear Neumann problems with concave and convex terms, Positivity, 16 (2012), 271-296. doi: 10.1007/s11117-011-0124-x. [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., A 35 (2015), 5003-5036. [31] V. Radulescu and D. Repovs, Combined effects in noninear problems arising in the study of anisotropic continuous media, Nonlinear Anal., 75 (2012), 1524-1530. doi: 10.1016/j.na.2011.01.037. [32] S. Takeuchi, Positive solutions of a degenerate elliptic equation with a logistic reaction, Proc. Amer. Math. Soc., 129 (2001), 433-441. doi: 10.1090/S0002-9939-00-05723-3. [33] S. Takeuchi, Multiplicity result for a degenerate elliptic equation with logistic reaction, J. Differential Equations, 173 (2001), 138-144. doi: 10.1006/jdeq.2000.3914. [34] S. Takeuchi and Y. Yamada, Asymptotic properties of a reaction-diffusion equation with a degenerate $p$-Laplacian, Nonlinear Anal., 42 (2000), 41-61. doi: 10.1016/S0362-546X(98)00329-0. [35] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.
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