• Previous Article
    Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity
  • CPAA Home
  • This Issue
  • Next Article
    Some results on two-dimensional Hénon equation with large exponent in nonlinearity
March  2013, 12(2): 815-829. doi: 10.3934/cpaa.2013.12.815

Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 6, 95125 Catania

2. 

Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received  September 2011 Revised  January 2012 Published  September 2012

A nonlinear elliptic equation with $p$-Laplacian, concave-convex reaction term depending on a parameter $\lambda>0$, and homogeneous boundary condition, is investigated. A bifurcation result, which describes the set of positive solutions as $\lambda$ varies, is obtained through variational methods combined with truncation and comparison techniques.
Citation: Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure and Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196 (2008).

[2]

A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[3]

D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplace operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447.

[4]

D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity, Bull. Austral. Math. Soc., 77 (2008), 285-303. doi: 10.1017/S0004972708000282.

[5]

L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents, Nonlinear Anal., 24 (1995), 1639-1648. doi: 10.1016/0362-546X(94)E0054-K.

[6]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the $p$-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 737-752. doi: 10.1017/S0308210509000845.

[7]

L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Ser. Math. Anal. Appl., 9, Chapman and Hall/CRC Press, Boca Raton, 2006.

[8]

L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Ser. Math. Anal. Appl., 8, Chapman and Hall/CRC Press, Boca Raton, 2005.

[9]

J. P. Garcia Azorero, J. 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.

[10]

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.

[11]

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.

[12]

An Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099. doi: 10.1016/j.na.2005.05.056.

[13]

S. Li, S. Wu and H.-S. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224. doi: 10.1006/jdeq.2001.4167.

[14]

G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic 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.

[15]

P. Lindqvist, On the equation div$(|\nabla u|^{p-2}\nabla u) +\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.1090/S0002-9939-1990-1007505-7.

[16]

O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz condition, J. Differential Equations, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035.

[17]

I. Peral, Some results on quasilinear elliptic equations: growth versus shape, in "Nonlinear Functional Analysis and Applications to Differential Equations (Trieste 1997)" (A. Ambrosetti, K.-C. Chang and I. Ekeland eds.), World Sci. Publ., River Edge, NJ, (1998), 153-202.

[18]

J. L. Vázquez, 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, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196 (2008).

[2]

A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[3]

D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplace operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447.

[4]

D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity, Bull. Austral. Math. Soc., 77 (2008), 285-303. doi: 10.1017/S0004972708000282.

[5]

L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents, Nonlinear Anal., 24 (1995), 1639-1648. doi: 10.1016/0362-546X(94)E0054-K.

[6]

G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the $p$-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 737-752. doi: 10.1017/S0308210509000845.

[7]

L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Ser. Math. Anal. Appl., 9, Chapman and Hall/CRC Press, Boca Raton, 2006.

[8]

L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Ser. Math. Anal. Appl., 8, Chapman and Hall/CRC Press, Boca Raton, 2005.

[9]

J. P. Garcia Azorero, J. 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.

[10]

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.

[11]

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.

[12]

An Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099. doi: 10.1016/j.na.2005.05.056.

[13]

S. Li, S. Wu and H.-S. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224. doi: 10.1006/jdeq.2001.4167.

[14]

G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic 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.

[15]

P. Lindqvist, On the equation div$(|\nabla u|^{p-2}\nabla u) +\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164. doi: 10.1090/S0002-9939-1990-1007505-7.

[16]

O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz condition, J. Differential Equations, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035.

[17]

I. Peral, Some results on quasilinear elliptic equations: growth versus shape, in "Nonlinear Functional Analysis and Applications to Differential Equations (Trieste 1997)" (A. Ambrosetti, K.-C. Chang and I. Ekeland eds.), World Sci. Publ., River Edge, NJ, (1998), 153-202.

[18]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.

[1]

Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107

[2]

Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036

[3]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076

[4]

M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3445-3479. doi: 10.3934/cpaa.2021113

[5]

Leszek Gasiński, Nikolaos S. Papageorgiou. Singular equations with variable exponents and concave-convex nonlinearities. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022135

[6]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922

[7]

Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073

[8]

Junping Shi, Ratnasingham Shivaji. Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 559-571. doi: 10.3934/dcds.2001.7.559

[9]

Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008

[10]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1031-1051. doi: 10.3934/cpaa.2009.8.1031

[11]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715

[12]

Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci. A nonlocal concave-convex problem with nonlocal mixed boundary data. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1103-1120. doi: 10.3934/cpaa.2018053

[13]

Shouchuan Hu, Nikolas S. Papageorgiou. Positive solutions for resonant (p, q)-equations with concave terms. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2639-2656. doi: 10.3934/cpaa.2018125

[14]

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

[15]

Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure and Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941

[16]

Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427

[17]

Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166

[18]

Leszek Gasiński, Nikolaos S. Papageorgiou. A pair of positive solutions for $(p,q)$-equations with combined nonlinearities. Communications on Pure and Applied Analysis, 2014, 13 (1) : 203-215. doi: 10.3934/cpaa.2014.13.203

[19]

João Marcos do Ó, Uberlandio Severo. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Communications on Pure and Applied Analysis, 2009, 8 (2) : 621-644. doi: 10.3934/cpaa.2009.8.621

[20]

Anna Maria Candela, Addolorata Salvatore. Positive solutions for some generalized $ p $–Laplacian type problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1935-1945. doi: 10.3934/dcdss.2020151

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (247)
  • HTML views (0)
  • Cited by (29)

[Back to Top]