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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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [8] J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent,, Portugaliae Mathematica, 57 (2000), 273.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, Comm. Math. Phys., 131 (1990), 223.   Google Scholar [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.  Google Scholar [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.  Google Scholar [20] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, vol IV,", Academic Press, (1978).   Google Scholar [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.  Google Scholar [22] J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth,, preprint, ().   Google Scholar [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.   Google Scholar

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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.  doi: 10.1007/s002050050067.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [8] J. Chabrowski and J. Yang, Multiple semilclassical solutions of the Schrödinger equation involving a critical Sobolev exponent,, Portugaliae Mathematica, 57 (2000), 273.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, Comm. Math. Phys., 131 (1990), 223.   Google Scholar [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.  Google Scholar [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.  Google Scholar [20] M. Reed and B. Simon, "Methods of Modern Mathematical Physics, vol IV,", Academic Press, (1978).   Google Scholar [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.  Google Scholar [22] J. Zhang, Z. Chen and W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth,, preprint, ().   Google Scholar [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.   Google Scholar
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