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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

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

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.
Citation: Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure & 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. 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. 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.

[4]

T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation,, Z. angew. Math. Phys., 51 (2000), 266. 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. 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. 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.

[8]

J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent,, Portugaliae Mathematica, 57 (2000), 273.

[9]

J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent,, Z. angew. Math. Phys., 49 (1998), 276. 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. 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. doi: 10.1017/S030821050002730X.

[12]

M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations,, Ann. Inst. Henri Poincar$\acutee$, 15 (1998), 127. 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. 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. 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. 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. 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.

[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. doi: 10.1080/03605308808820585.

[19]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals,, Milan J.Math., 73 (2005), 563. 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. 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, ().

[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.

show all references

References:
[1]

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, Arch. Ration. Mech. Anal., 140 (1997), 285. 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. 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.

[4]

T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation,, Z. angew. Math. Phys., 51 (2000), 266. 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. 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. 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.

[8]

J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent,, Portugaliae Mathematica, 57 (2000), 273.

[9]

J. Chabrowski and J. Yang, Existence theorems for the Schrödinger equation involving a critical Sobolev exponent,, Z. angew. Math. Phys., 49 (1998), 276. 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. 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. doi: 10.1017/S030821050002730X.

[12]

M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations,, Ann. Inst. Henri Poincar$\acutee$, 15 (1998), 127. 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. 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. 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. 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. 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.

[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. doi: 10.1080/03605308808820585.

[19]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals,, Milan J.Math., 73 (2005), 563. 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. 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, ().

[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.

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