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

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

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

Yi-hsin Cheng ^1, and Tsung-Fang Wu ^1,

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

Received March 2016 Revised June 2016 Published September 2016

Abstract
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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.

Keywords: concave-convex nonlinearities, steep potential., Semilinear elliptic problems, variational methods.

Mathematics Subject Classification: Primary: 35J20; Secondary: 35J6.

Citation: Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure and 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, J. Funct. Anal., 137 (1996), 219-242. doi: 10.1006/jfan.1996.0045.
[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]	Adimurthy, F. Pacella and L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball, Diff. Int. Equations, 10 (1997), 1157-1170.
[4]	P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electr. J. Diff. Eqns., 5 (1997), 1-11.
[5]	T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494.
[6]	T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N2$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149.
[7]	K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electr. J. Diff. Eqns., 69 (2007), 1-9.
[8]	K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications, Diff. Int. Equations, 22 (2009), 1097-1114.
[9]	K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equns, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.
[10]	J. Chabrowski and João Marcos 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.
[11]	C. Y. Chen and T. F. Wu, Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent, Proc. Roy. Soc. Edinburgh Sect., 144A (2014), 691-709. doi: 10.1017/S0308210512000133.
[12]	L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Annls Inst. H. Poincaré Analyse Non linéaire, 16 (1999), 631-652. doi: 10.1016/S0294-1449(99)80030-4.
[13]	P. Drábek and S. I. Pohozaev, Positive solutions for the $p$ -Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.
[14]	Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419. doi: 10.1007/s00526-006-0071-8.
[15]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.
[16]	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.
[17]	J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^N2$ involving subcritical exponents, Nonlinear Analysis: T. M. A., 32 (1998), 41-51. doi: 10.1016/S0362-546X(97)00451-3.
[18]	T. S. Hsu and H. L. Lin, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^N2$ involving concave-convex nonlineatlties and sign-changing weight functions, Abstract and Applied Analysis, 2010 ID658397 (2010), 21 pages.
[19]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case Part I, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.
[20]	F. F. Liao and C. L. Tang, Four positive solutions of a quasilinear elliptic equation in $\mathbb{R}^N2$, Comm. Pure Appl. Anal., 12 (2013), 2577-2600. doi: 10.3934/cpaa.2013.12.2577.
[21]	Z. Liu and Z. Q. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys.,56 (2005), 609-629. doi: 10.1007/s00033-005-3115-6.
[22]	T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem II, J. Diff. Eqns., 158 (1999), 94-151. doi: 10.1016/S0022-0396(99)80020-5.
[23]	Francisco Odair de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity, J. Func. Anal., 261 (2011), 2569-2586. doi: 10.1016/j.jfa.2011.07.002.
[24]	M. Struwe, Variational Methods, 2nd edition, Springer-Verlag, Berlin-Heidelberg, 1996. doi: 10.1007/978-3-662-03212-1.
[25]	M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 705-717. doi: 10.1017/S0308210500002614.
[26]	T. F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in $\mathbb{R}^N2$, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 647-670. doi: 10.1017/S0308210506001156.
[27]	T. F. Wu, Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight, J. Differ. Equat., 249 (2010), 1459-1578. doi: 10.1016/j.jde.2010.07.021.
[28]	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.
[29]	T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problem in $\mathbb{R}^N2$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131. doi: 10.1016/j.jfa.2009.08.005.
[30]	H. Yin, Z. Yang and Z. Feng, Multiple positive solutions for a quasilinear elliptic equation in $\mathbb{R}^N2$, Diff. Integ. Eqns, 25 (2012), 977-992.
[31]	L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Diff. Eqns., 255 (2013), 1-23. doi: 10.1016/j.jde.2013.03.005.

show all references

References:

[1]	A. Ambrosetti, G. J. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242. doi: 10.1006/jfan.1996.0045.
[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]	Adimurthy, F. Pacella and L. Yadava, On the number of positive solutions of some semilinear Dirichlet problems in a ball, Diff. Int. Equations, 10 (1997), 1157-1170.
[4]	P. A. Binding, P. Drábek and Y. X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electr. J. Diff. Eqns., 5 (1997), 1-11.
[5]	T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494.
[6]	T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N2$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149.
[7]	K. J. Brown and T. F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electr. J. Diff. Eqns., 69 (2007), 1-9.
[8]	K. J. Brown and T. F. Wu, A fibering map approach to a potential operator equation and its applications, Diff. Int. Equations, 22 (2009), 1097-1114.
[9]	K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equns, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.
[10]	J. Chabrowski and João Marcos 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.
[11]	C. Y. Chen and T. F. Wu, Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent, Proc. Roy. Soc. Edinburgh Sect., 144A (2014), 691-709. doi: 10.1017/S0308210512000133.
[12]	L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Annls Inst. H. Poincaré Analyse Non linéaire, 16 (1999), 631-652. doi: 10.1016/S0294-1449(99)80030-4.
[13]	P. Drábek and S. I. Pohozaev, Positive solutions for the $p$ -Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.
[14]	Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419. doi: 10.1007/s00526-006-0071-8.
[15]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353.
[16]	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.
[17]	J. V. Goncalves and O. H. Miyagaki, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^N2$ involving subcritical exponents, Nonlinear Analysis: T. M. A., 32 (1998), 41-51. doi: 10.1016/S0362-546X(97)00451-3.
[18]	T. S. Hsu and H. L. Lin, Multiple positive solutions for semilinear elliptic equations in $\mathbb{R}^N2$ involving concave-convex nonlineatlties and sign-changing weight functions, Abstract and Applied Analysis, 2010 ID658397 (2010), 21 pages.
[19]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case Part I, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 109-145.
[20]	F. F. Liao and C. L. Tang, Four positive solutions of a quasilinear elliptic equation in $\mathbb{R}^N2$, Comm. Pure Appl. Anal., 12 (2013), 2577-2600. doi: 10.3934/cpaa.2013.12.2577.
[21]	Z. Liu and Z. Q. Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys.,56 (2005), 609-629. doi: 10.1007/s00033-005-3115-6.
[22]	T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problem II, J. Diff. Eqns., 158 (1999), 94-151. doi: 10.1016/S0022-0396(99)80020-5.
[23]	Francisco Odair de Paiva, Nonnegative solutions of elliptic problems with sublinear indefinite nonlinearity, J. Func. Anal., 261 (2011), 2569-2586. doi: 10.1016/j.jfa.2011.07.002.
[24]	M. Struwe, Variational Methods, 2nd edition, Springer-Verlag, Berlin-Heidelberg, 1996. doi: 10.1007/978-3-662-03212-1.
[25]	M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 705-717. doi: 10.1017/S0308210500002614.
[26]	T. F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in $\mathbb{R}^N2$, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 647-670. doi: 10.1017/S0308210506001156.
[27]	T. F. Wu, Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight, J. Differ. Equat., 249 (2010), 1459-1578. doi: 10.1016/j.jde.2010.07.021.
[28]	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.
[29]	T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problem in $\mathbb{R}^N2$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99-131. doi: 10.1016/j.jfa.2009.08.005.
[30]	H. Yin, Z. Yang and Z. Feng, Multiple positive solutions for a quasilinear elliptic equation in $\mathbb{R}^N2$, Diff. Integ. Eqns, 25 (2012), 977-992.
[31]	L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Diff. Eqns., 255 (2013), 1-23. doi: 10.1016/j.jde.2013.03.005.

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