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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 $R^N$

 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 and 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. [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. [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. [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. [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. [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. [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. [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. [9] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [9] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. [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. [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. [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. [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. [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. [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.
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