Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials

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January 2014, 13(1): 237-248. doi: 10.3934/cpaa.2014.13.237

Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials

Zhongwei Tang ^1,

1.	School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

Received October 2012 Revised May 2013 Published July 2013

Abstract
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In this paper, we consider the following semilinear Schrödinger equations with ciritical growth \begin{eqnarray} -\Delta u+(\lambda a(x)-\delta)u=|u|^{2^*-2}u,x\in R^N, \end{eqnarray} where $N\geq 4$, $a(x)\geq 0$ and its zero sets are not empty. $2^*$ is the critical Sobolev exponent. $\delta>0$ is a constant such that the operator $-\Delta +\lambda a(x)-\delta$ might be indefinite but is non-degenerate. We prove the existence of least energy solutions which localize near the potential well $int \{a^{-1}(0)\}$ for $\lambda$ large enough.

Keywords: Semilinear Schrödinger equation, critical Sobolev exponent, least energy solutions, potential well..

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J6.

Citation: Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure and Applied Analysis, 2014, 13 (1) : 237-248. doi: 10.3934/cpaa.2014.13.237

References:

[1]	A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.
[2]	A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.
[3]	T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential, Discrete Continuous Dynam. Systems, 33 (2013), 7-26.
[4]	T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. angew. Math. Phys., 51 (2000), 266-284. doi: 10.1007/s000330050003.
[5]	V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{\frac{N+2}{N-2}}$ in $R^N$, J. Funct. Anal., 88 (1990), 90-117. doi: 10.1016/0022-1236(90)90120-A.
[6]	J. Byeon and Z. Q. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II, Calc.Var., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8.
[7]	A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, 2 (1985), 463-470.
[8]	J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugaliae Mathematica, 57 (2000), 273-284.
[9]	J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent, Z. angew. Math. Phys., 49 (1998), 276-293. doi: 10.1007/PL00001485.
[10]	S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff. Equat., 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662.
[11]	S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Royal Soc. Edinburgh, 128 (1998), 1249-1260. doi: 10.1017/S030821050002730X.
[12]	M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincar$é$, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7.
[13]	M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.
[14]	Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal., 251 (2007), 546-572. doi: 10.1016/j.jfa.2007.07.005.
[15]	A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.
[16]	C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Part. Diff. Equat., 21 (1996), 787-820. doi: 10.1080/03605309608821208.
[17]	Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.
[18]	Y.-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $(V)_a$, Comm. Part. Diff. Equat., 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.
[19]	A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J.Math., 73 (2005), 563-574. doi: 10.1007/s00032-005-0047-8.
[20]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics, vol IV," Academic Press, 1978.
[21]	A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.
[22]	J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, preprint, arXiv:1209.3074.
[23]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I, Ann.Inst.H.Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

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References:

[1]	A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067.
[2]	A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271. doi: 10.1007/s002050100152.
[3]	T. Bartsch and Z. Tang, Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential, Discrete Continuous Dynam. Systems, 33 (2013), 7-26.
[4]	T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. angew. Math. Phys., 51 (2000), 266-284. doi: 10.1007/s000330050003.
[5]	V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{\frac{N+2}{N-2}}$ in $R^N$, J. Funct. Anal., 88 (1990), 90-117. doi: 10.1016/0022-1236(90)90120-A.
[6]	J. Byeon and Z. Q. Wang, Standing waves with a ciritical frequency for nonlinear Schrödinger equations II, Calc.Var., 18 (2003), 207-219. doi: 10.1007/s00526-002-0191-8.
[7]	A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, 2 (1985), 463-470.
[8]	J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent, Portugaliae Mathematica, 57 (2000), 273-284.
[9]	J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent, Z. angew. Math. Phys., 49 (1998), 276-293. doi: 10.1007/PL00001485.
[10]	S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff. Equat., 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662.
[11]	S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Royal Soc. Edinburgh, 128 (1998), 1249-1260. doi: 10.1017/S030821050002730X.
[12]	M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincar$é$, 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7.
[13]	M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.
[14]	Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal., 251 (2007), 546-572. doi: 10.1016/j.jfa.2007.07.005.
[15]	A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.
[16]	C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Part. Diff. Equat., 21 (1996), 787-820. doi: 10.1080/03605309608821208.
[17]	Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.
[18]	Y.-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class $(V)_a$, Comm. Part. Diff. Equat., 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.
[19]	A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J.Math., 73 (2005), 563-574. doi: 10.1007/s00032-005-0047-8.
[20]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics, vol IV," Academic Press, 1978.
[21]	A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.
[22]	J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, preprint, arXiv:1209.3074.
[23]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I, Ann.Inst.H.Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

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