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
References:
[1]

A. Ambrosetti, G. J. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations,, \emph{J. Funct. Anal.}, 137 (1996), 219. doi: 10.1006/jfan.1996.0045. Google Scholar

[2]

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

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Adimurthy, F. Pacella and L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball,, \emph{Diff. Int. Equations}, 10 (1997), 1157. Google Scholar

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P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations,, \emph{Electr. J. Diff. Eqns.}, 5 (1997), 1. Google Scholar

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T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well,, \emph{Commun. Contemp. Math.}, 3 (2001), 549. doi: 10.1142/S0219199701000494. Google Scholar

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K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications,, \emph{Diff. Int. Equations}, 22 (2009), 1097. Google Scholar

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K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,, \emph{J. Diff. Equns}, 193 (2003), 481. doi: 10.1016/S0022-0396(03)00121-9. Google Scholar

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

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L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle,, \emph{Annls Inst. H. Poincar\'e Analyse Non lin\'eaire}, 16 (1999), 631. doi: 10.1016/S0294-1449(99)80030-4. Google Scholar

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P. Drábek and S. I. Pohozaev, Positive solutions for the $p$ -Laplacian: application of the fibering method,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 127 (1997), 703. doi: 10.1017/S0308210500023787. Google Scholar

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Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential,, \emph{Calc. Var. Partial Differential Equations}, 29 (2007), 397. doi: 10.1007/s00526-006-0071-8. Google Scholar

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

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J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbbR^N$ involving subcritical exponents,, \emph{Nonlinear Analysis: T. M. A.}, 32 (1998), 41. doi: 10.1016/S0362-546X(97)00451-3. Google Scholar

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T. S. Hsu and H. L. Lin, Multiple positive solutions for semilinear elliptic equations in $\mathbbR^N$ involving concave-convex nonlineatlties and sign-changing weight functions,, \emph{Abstract and Applied Analysis}, 2010 (2010). Google Scholar

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P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case Part I,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lineair\'e}, 1 (1984), 109. Google Scholar

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F. F. Liao and C. L. Tang, Four positive solutions of a quasilinear elliptic equation in $\mathbbR^N$,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 2577. doi: 10.3934/cpaa.2013.12.2577. Google Scholar

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Z. Liu and Z. Q. Wang, Schrödinger equations with concave and convex nonlinearities,, \emph{Z. Angew. Math. Phys.}, 56 (2005), 609. doi: 10.1007/s00033-005-3115-6. Google Scholar

[22]

T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem II,, \emph{J. Diff. Eqns.}, 158 (1999), 94. doi: 10.1016/S0022-0396(99)80020-5. Google Scholar

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Francisco Odair de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity,, \emph{J. Func. Anal.}, 261 (2011), 2569. doi: 10.1016/j.jfa.2011.07.002. Google Scholar

[24]

M. Struwe, Variational Methods,, 2$^{nd}$ edition, (1996). doi: 10.1007/978-3-662-03212-1. Google Scholar

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M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 133 (2003), 705. doi: 10.1017/S0308210500002614. Google Scholar

[26]

T. F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in $\mathbbR^N$,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138 (2008), 647. doi: 10.1017/S0308210506001156. Google Scholar

[27]

T. F. Wu, Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight,, \emph{J. Differ. Equat.}, 249 (2010), 1459. doi: 10.1016/j.jde.2010.07.021. Google Scholar

[28]

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

[29]

T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problem in $\mathbbR^N$ involving sign-changing weight,, \emph{J. Funct. Anal.}, 258 (2010), 99. doi: 10.1016/j.jfa.2009.08.005. Google Scholar

[30]

H. Yin, Z. Yang and Z. Feng, Multiple positive solutions for a quasilinear elliptic equation in $\mathbbR^N$,, \emph{Diff. Integ. Eqns}, 25 (2012), 977. Google Scholar

[31]

L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential,, \emph{J. Diff. Eqns.}, 255 (2013), 1. doi: 10.1016/j.jde.2013.03.005. Google Scholar

show all references

References:
[1]

A. Ambrosetti, G. J. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations,, \emph{J. Funct. Anal.}, 137 (1996), 219. doi: 10.1006/jfan.1996.0045. Google Scholar

[2]

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

[3]

Adimurthy, F. Pacella and L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball,, \emph{Diff. Int. Equations}, 10 (1997), 1157. Google Scholar

[4]

P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations,, \emph{Electr. J. Diff. Eqns.}, 5 (1997), 1. Google Scholar

[5]

T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well,, \emph{Commun. Contemp. Math.}, 3 (2001), 549. doi: 10.1142/S0219199701000494. Google Scholar

[6]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbbR^N$,, \emph{Comm. Partial Differential Equations}, 20 (1995), 1725. doi: 10.1080/03605309508821149. Google Scholar

[7]

K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem,, \emph{Electr. J. Diff. Eqns.}, 69 (2007), 1. Google Scholar

[8]

K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications,, \emph{Diff. Int. Equations}, 22 (2009), 1097. Google Scholar

[9]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,, \emph{J. Diff. Equns}, 193 (2003), 481. doi: 10.1016/S0022-0396(03)00121-9. Google Scholar

[10]

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

[11]

C. Y. Chen and T. F. Wu, Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent,, \emph{Proc. Roy. Soc. Edinburgh Sect.}, 144A (2014), 691. doi: 10.1017/S0308210512000133. Google Scholar

[12]

L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle,, \emph{Annls Inst. H. Poincar\'e Analyse Non lin\'eaire}, 16 (1999), 631. doi: 10.1016/S0294-1449(99)80030-4. Google Scholar

[13]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p$ -Laplacian: application of the fibering method,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 127 (1997), 703. doi: 10.1017/S0308210500023787. Google Scholar

[14]

Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential,, \emph{Calc. Var. Partial Differential Equations}, 29 (2007), 397. doi: 10.1007/s00526-006-0071-8. Google Scholar

[15]

I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 17 (1974), 324. Google Scholar

[16]

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

[17]

J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbbR^N$ involving subcritical exponents,, \emph{Nonlinear Analysis: T. M. A.}, 32 (1998), 41. doi: 10.1016/S0362-546X(97)00451-3. Google Scholar

[18]

T. S. Hsu and H. L. Lin, Multiple positive solutions for semilinear elliptic equations in $\mathbbR^N$ involving concave-convex nonlineatlties and sign-changing weight functions,, \emph{Abstract and Applied Analysis}, 2010 (2010). Google Scholar

[19]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case Part I,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lineair\'e}, 1 (1984), 109. Google Scholar

[20]

F. F. Liao and C. L. Tang, Four positive solutions of a quasilinear elliptic equation in $\mathbbR^N$,, \emph{Comm. Pure Appl. Anal.}, 12 (2013), 2577. doi: 10.3934/cpaa.2013.12.2577. Google Scholar

[21]

Z. Liu and Z. Q. Wang, Schrödinger equations with concave and convex nonlinearities,, \emph{Z. Angew. Math. Phys.}, 56 (2005), 609. doi: 10.1007/s00033-005-3115-6. Google Scholar

[22]

T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem II,, \emph{J. Diff. Eqns.}, 158 (1999), 94. doi: 10.1016/S0022-0396(99)80020-5. Google Scholar

[23]

Francisco Odair de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity,, \emph{J. Func. Anal.}, 261 (2011), 2569. doi: 10.1016/j.jfa.2011.07.002. Google Scholar

[24]

M. Struwe, Variational Methods,, 2$^{nd}$ edition, (1996). doi: 10.1007/978-3-662-03212-1. Google Scholar

[25]

M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 133 (2003), 705. doi: 10.1017/S0308210500002614. Google Scholar

[26]

T. F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in $\mathbbR^N$,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138 (2008), 647. doi: 10.1017/S0308210506001156. Google Scholar

[27]

T. F. Wu, Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight,, \emph{J. Differ. Equat.}, 249 (2010), 1459. doi: 10.1016/j.jde.2010.07.021. Google Scholar

[28]

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

[29]

T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problem in $\mathbbR^N$ involving sign-changing weight,, \emph{J. Funct. Anal.}, 258 (2010), 99. doi: 10.1016/j.jfa.2009.08.005. Google Scholar

[30]

H. Yin, Z. Yang and Z. Feng, Multiple positive solutions for a quasilinear elliptic equation in $\mathbbR^N$,, \emph{Diff. Integ. Eqns}, 25 (2012), 977. Google Scholar

[31]

L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential,, \emph{J. Diff. Eqns.}, 255 (2013), 1. doi: 10.1016/j.jde.2013.03.005. Google Scholar

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