Positive solutions for  parametric $p$-Laplacian equations

Nikolaos S. Papageorgiou; George  Smyrlis

doi:10.3934/cpaa.2016002

Previous Article
Scattering for a nonlinear Schrödinger equation with a potential
CPAA Home
This Issue
Next Article
Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers

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

Positive solutions for parametric $p$-Laplacian equations

Nikolaos S. Papageorgiou ^1, and George Smyrlis ^2,

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

Abstract
Related Papers

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.

Keywords: strong comparison principle, $p$-logistic equations with competing nonlinearities, superlinear perturbation., bifurcation-type theorem.

Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35J9.

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

[1]	Xin-Guang Yang, Marcelo J. D. Nascimento, Maurício L. Pelicer. Uniform attractors for non-autonomous plate equations with $ p $-Laplacian perturbation and critical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1937-1961. doi: 10.3934/dcds.2020100
[2]	Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5003-5036. doi: 10.3934/dcds.2015.35.5003
[3]	Alexander Krasnosel'skii, Alexei Pokrovskii. On subharmonics bifurcation in equations with homogeneous nonlinearities. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 747-762. doi: 10.3934/dcds.2001.7.747
[4]	Michael E. Filippakis, Donal O'Regan, Nikolaos S. Papageorgiou. Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1507-1527. doi: 10.3934/cpaa.2010.9.1507
[5]	Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017
[6]	Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725
[7]	Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
[8]	Yijing Sun, Yuxin Tan. Kirchhoff type equations with strong singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 181-193. doi: 10.3934/cpaa.2019010
[9]	Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[10]	Pawan Kumar Mishra, Sarika Goyal, K. Sreenadh. Polyharmonic Kirchhoff type equations with singular exponential nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1689-1717. doi: 10.3934/cpaa.2016009
[11]	Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555
[12]	Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[13]	Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1031-1051. doi: 10.3934/cpaa.2009.8.1031
[14]	Leszek Gasiński, Nikolaos S. Papageorgiou. A pair of positive solutions for $(p,q)$-equations with combined nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 203-215. doi: 10.3934/cpaa.2014.13.203
[15]	A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003
[16]	Santiago Cano-Casanova. Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions. Conference Publications, 2013, 2013 (special) : 95-104. doi: 10.3934/proc.2013.2013.95
[17]	Inácio Andruski-Guimarães, Anselmo Chaves-Neto. Estimation in polytomous logistic model: Comparison of methods. Journal of Industrial & Management Optimization, 2009, 5 (2) : 239-252. doi: 10.3934/jimo.2009.5.239
[18]	Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296
[19]	Nakao Hayashi, Pavel I. Naumkin. Modified wave operator for Schrodinger type equations with subcritical dissipative nonlinearities. Inverse Problems & Imaging, 2007, 1 (2) : 391-398. doi: 10.3934/ipi.2007.1.391
[20]	M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences

Positive solutions for parametric $p$-Laplacian equations

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Positive solutions for parametric $p$-Laplacian equations

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors