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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.

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, 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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