American Institute of Mathematical Sciences

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

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

Fang-Fang Liao 1, and  Chun-Lei Tang 1,

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.
Keywords: positive solutions., Nehari manifold, $p-$Laplacian, Palais-Smale decomposition lemma.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J62; 35J9.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[1]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[2]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[3]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[4]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[5]

M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403
[6]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[7]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109
[8]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
[9]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
[10]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[11]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[12]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
[13]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[14]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069
[15]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[16]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[17]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[18]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[19]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378
[20]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences