November  2016, 15(6): 2457-2473. doi: 10.3934/cpaa.2016044

Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential

1. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

Received  March 2016 Revised  June 2016 Published  September 2016

In this paper, we study the existence, multiplicity and concentration of positive solutions for the following indefinite semilinear elliptic equations involving concave-convex nonlinearities: \begin{eqnarray} \left\{\begin{array}{l} -\Delta u+V_{\lambda }\left( x\right) u=f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, \\ u\geq 0, & \text{in }\mathbb{R}^{N}, \end{array}\right. \end{eqnarray} where$ 1 < q < 2 < p < 2^{\ast} (2^{\ast } = \frac{2N}{N-2}$ for $N \geq 3) $ the potential $V_{\lambda }(x)=\lambda V^{+}(x)-V^{-}(x)$ with $ V^{\pm }=\max \left\{ \pm V,0\right\} $ and the parameter $\lambda >0.$ We assume that the functions $f,g$ and $V$ satisfy suitable conditions with the potential $V$ and the weight function $g$ without the assumptions of infinite limits.
Citation: Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044
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show all references

References:
[1]

J. Funct. Anal., 137 (1996), 219-242. doi: 10.1006/jfan.1996.0045.  Google Scholar

[2]

J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.  Google Scholar

[3]

Diff. Int. Equations, 10 (1997), 1157-1170.  Google Scholar

[4]

Electr. J. Diff. Eqns., 5 (1997), 1-11.  Google Scholar

[5]

Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494.  Google Scholar

[6]

Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149.  Google Scholar

[7]

Electr. J. Diff. Eqns., 69 (2007), 1-9.  Google Scholar

[8]

Diff. Int. Equations, 22 (2009), 1097-1114.  Google Scholar

[9]

J. Diff. Equns, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[10]

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

[11]

Proc. Roy. Soc. Edinburgh Sect., 144A (2014), 691-709. doi: 10.1017/S0308210512000133.  Google Scholar

[12]

Annls Inst. H. Poincaré Analyse Non linéaire, 16 (1999), 631-652. doi: 10.1016/S0294-1449(99)80030-4.  Google Scholar

[13]

Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.  Google Scholar

[14]

Calc. Var. Partial Differential Equations, 29 (2007), 397-419. doi: 10.1007/s00526-006-0071-8.  Google Scholar

[15]

J. Math. Anal. Appl., 17 (1974), 324-353.  Google Scholar

[16]

J. Funct. Anal., 199 (2003), 452-467. doi: 10.1016/S0022-1236(02)00060-5.  Google Scholar

[17]

Nonlinear Analysis: T. M. A., 32 (1998), 41-51. doi: 10.1016/S0362-546X(97)00451-3.  Google Scholar

[18]

Abstract and Applied Analysis, 2010 ID658397 (2010), 21 pages.  Google Scholar

[19]

Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.  Google Scholar

[20]

Comm. Pure Appl. Anal., 12 (2013), 2577-2600. doi: 10.3934/cpaa.2013.12.2577.  Google Scholar

[21]

Z. Angew. Math. Phys.,56 (2005), 609-629. doi: 10.1007/s00033-005-3115-6.  Google Scholar

[22]

J. Diff. Eqns., 158 (1999), 94-151. doi: 10.1016/S0022-0396(99)80020-5.  Google Scholar

[23]

J. Func. Anal., 261 (2011), 2569-2586. doi: 10.1016/j.jfa.2011.07.002.  Google Scholar

[24]

2nd edition, Springer-Verlag, Berlin-Heidelberg, 1996. doi: 10.1007/978-3-662-03212-1.  Google Scholar

[25]

Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 705-717. doi: 10.1017/S0308210500002614.  Google Scholar

[26]

Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 647-670. doi: 10.1017/S0308210506001156.  Google Scholar

[27]

J. Differ. Equat., 249 (2010), 1459-1578. doi: 10.1016/j.jde.2010.07.021.  Google Scholar

[28]

J. Math. Anal. Appl., 318 (2006), 253-270. doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar

[29]

J. Funct. Anal., 258 (2010), 99-131. doi: 10.1016/j.jfa.2009.08.005.  Google Scholar

[30]

Diff. Integ. Eqns, 25 (2012), 977-992.  Google Scholar

[31]

J. Diff. Eqns., 255 (2013), 1-23. doi: 10.1016/j.jde.2013.03.005.  Google Scholar

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