# American Institute of Mathematical Sciences

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 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. 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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 $\mathbbR^N$ 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 $\mathbbR^N$ 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 $\mathbbR^N$, 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 $\mathbbR^N$, 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 $\mathbbR^N$ 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 $\mathbbR^N$, 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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##### 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 $\mathbbR^N$, 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 $\mathbbR^N$ 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 $\mathbbR^N$ 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 $\mathbbR^N$, 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 $\mathbbR^N$, 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 $\mathbbR^N$ 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 $\mathbbR^N$, 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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