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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), 63-95. doi: 10.1007/s005260050003.  Google Scholar

[2]

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[3]

M. Badiale and G. Citti, Concentration compactness principle and quasilinear elliptic equations in $R^n$, Comm. Partial Differential Equations, 16 (1991), 1795-1818. 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), 486-490. 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), 55-76. doi: 10.1002/1522-2616(200201)233:1<55::AID-MANA55>3.3.CO;2-I.  Google Scholar

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K.-C. Chang, "Infinite-dimensional Morse Theory and Multiple Solution Problems,'' Progress in Nonlinear Differential Equations and their Applications, 6. Birkhäuser Boston Inc., Boston, MA, 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), 291-308. 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), 452-467. doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[9]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  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), 947-974. 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), 758-775. 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), 2005-2012 (electronic). 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), 253-270. 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), 557-563.  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), 99-131. 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), 63-95. 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), 519-543. 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), 1795-1818. 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), 486-490. 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), 55-76. 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, 6. Birkhäuser Boston Inc., Boston, MA, 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), 291-308. 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), 452-467. doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[9]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  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), 947-974. 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), 758-775. 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), 2005-2012 (electronic). 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), 253-270. 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), 557-563.  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), 99-131. doi: 10.1016/j.jfa.2009.08.005.  Google Scholar

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