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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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