• Previous Article
    Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation
  • CPAA Home
  • This Issue
  • Next Article
    Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications
2013, 12(6): 2577-2600. doi: 10.3934/cpaa.2013.12.2577

Four positive solutions of a quasilinear elliptic equation in $ R^N$

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  July 2012 Revised  October 2012 Published  May 2013

This paper deals with the existence of multiple positive solutions of a quasilinear elliptic equation \begin{eqnarray} -\Delta_p u+u^{p-1} = a(x)u^{q-1}+\lambda h(x) u^{r-1}, \text{in} R^N; \\ u\geq 0, \text{ a.e. }x \in R^N;\\ u \in W^{1,p}(R^N), \end{eqnarray} where $1 < p \leq 2$, $N>p$ and $1 < r < p$ $< q < p^* ( = \frac{pN}{N-p})$. A Nehari manifold is defined by a $C^1-$functional $I$ and is decomposed into two parts. Our work is to find four positive solutions of Eq. (1) when parameter $\lambda$ is sufficiently small.
Citation: Fang-Fang Liao, Chun-Lei Tang. Four positive solutions of a quasilinear elliptic equation in $ R^N$. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2577-2600. doi: 10.3934/cpaa.2013.12.2577
References:
[1]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^p+f(x)$ in $R^N$,, Calc. Var. Partial Differential Equations, 11 (2000), 63. doi: 10.1007/s005260050003.

[2]

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

[3]

M. Badiale and G. Citti, Concentration compactness principle and quasilinear elliptic equations in $R^n$,, Comm. Partial Differential Equations, 16 (1991), 1795. doi: 10.1080/03605309108820823.

[4]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.2307/2044999.

[5]

J. Chabrowski and J. M. Bezzera do Ó, On semilinear elliptic equations involving concave and convex nonlinearities,, Math. Nachr., 233/234 (2002), 55. doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.3.CO;2-I.

[6]

K.-C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems,'', Progress in Nonlinear Differential Equations and their Applications, (1993).

[7]

L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of $p$-Laplace equations via the moving plane method,, Arch. Ration. Mech. Anal., 148 (1999), 291. doi: 10.1007/s002050050163.

[8]

D. G. De Figueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems,, J. Funct. Anal., 199 (2003), 452. doi: 10.1016/S0022-1236(02)00060-5.

[9]

I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324.

[10]

F. Gazzola, B. Peletier, P. Pucci and J. Serrin, Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. {II},, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 947. doi: 10.1016/S0294-1449(03)00013-1.

[11]

T.-S. Hsu and H.-L. Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $R^N$,, J. Math. Anal. Appl., 365 (2010), 758. doi: 10.1016/j.jmaa.2009.12.004.

[12]

Y. Li and C. Zhao, A note on exponential decay properties of ground states for quasilinear elliptic equations,, Proc. Amer. Math. Soc., 133 (2005), 2005. doi: 10.1090/S0002-9939-05-07870-6.

[13]

T.-F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,, J. Math. Anal. Appl., 318 (2006), 253. doi: 10.1016/j.jmaa.2005.05.057.

[14]

T.-F. Wu, Multiplicity of positive solution of $p$-Laplacian problems with sign-changing weight functions,, Int. J. Math. Anal. (Ruse), 1 (2007), 557.

[15]

T.-F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $R^N$ involving sign-changing weight,, J. Funct. Anal., 258 (2010), 99. doi: 10.1016/j.jfa.2009.08.005.

show all references

References:
[1]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^p+f(x)$ in $R^N$,, Calc. Var. Partial Differential Equations, 11 (2000), 63. doi: 10.1007/s005260050003.

[2]

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

[3]

M. Badiale and G. Citti, Concentration compactness principle and quasilinear elliptic equations in $R^n$,, Comm. Partial Differential Equations, 16 (1991), 1795. doi: 10.1080/03605309108820823.

[4]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486. doi: 10.2307/2044999.

[5]

J. Chabrowski and J. M. Bezzera do Ó, On semilinear elliptic equations involving concave and convex nonlinearities,, Math. Nachr., 233/234 (2002), 55. doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.3.CO;2-I.

[6]

K.-C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems,'', Progress in Nonlinear Differential Equations and their Applications, (1993).

[7]

L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of $p$-Laplace equations via the moving plane method,, Arch. Ration. Mech. Anal., 148 (1999), 291. doi: 10.1007/s002050050163.

[8]

D. G. De Figueiredo, J.-P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems,, J. Funct. Anal., 199 (2003), 452. doi: 10.1016/S0022-1236(02)00060-5.

[9]

I. Ekeland, On the variational principle,, J. Math. Anal. Appl., 47 (1974), 324.

[10]

F. Gazzola, B. Peletier, P. Pucci and J. Serrin, Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. {II},, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 947. doi: 10.1016/S0294-1449(03)00013-1.

[11]

T.-S. Hsu and H.-L. Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $R^N$,, J. Math. Anal. Appl., 365 (2010), 758. doi: 10.1016/j.jmaa.2009.12.004.

[12]

Y. Li and C. Zhao, A note on exponential decay properties of ground states for quasilinear elliptic equations,, Proc. Amer. Math. Soc., 133 (2005), 2005. doi: 10.1090/S0002-9939-05-07870-6.

[13]

T.-F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,, J. Math. Anal. Appl., 318 (2006), 253. doi: 10.1016/j.jmaa.2005.05.057.

[14]

T.-F. Wu, Multiplicity of positive solution of $p$-Laplacian problems with sign-changing weight functions,, Int. J. Math. Anal. (Ruse), 1 (2007), 557.

[15]

T.-F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $R^N$ involving sign-changing weight,, J. Funct. Anal., 258 (2010), 99. doi: 10.1016/j.jfa.2009.08.005.

[1]

Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17

[2]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829

[3]

Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289

[4]

A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987

[5]

Xianling Fan, Yuanzhang Zhao, Guifang Huang. Existence of solutions for the $p-$Laplacian with crossing nonlinearity. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1019-1024. doi: 10.3934/dcds.2002.8.1019

[6]

Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108

[7]

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

[8]

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

[9]

Leandro M. Del Pezzo, Julio D. Rossi. Eigenvalues for a nonlocal pseudo $p-$Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6737-6765. doi: 10.3934/dcds.2016093

[10]

Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813

[11]

Sergiu Aizicovici, Nikolaos S. Papageorgiou, V. Staicu. The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 431-456. doi: 10.3934/dcds.2009.25.431

[12]

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

[13]

Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063

[14]

Shanming Ji, Jingxue Yin, Yutian Li. Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2411-2439. doi: 10.3934/dcds.2018100

[15]

Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623

[16]

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

[17]

Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675

[18]

Guowei Dai, Ruyun Ma. Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 99-116. doi: 10.3934/dcds.2015.35.99

[19]

Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857

[20]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]