Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells

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July 2016, 15(4): 1215-1231. doi: 10.3934/cpaa.2016.15.1215

Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells

Miaomiao Niu ^1, and Zhongwei Tang ^1,

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

Received June 2015 Revised January 2016 Published April 2016

Abstract
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In this paper, we are concerned with the existence of least energy solutions of nonlinear Schrödinger equations involving the half Laplacian \begin{eqnarray} (-\Delta)^{1/2}u(x)+\lambda V(x)u(x)=u(x)^{p-1}, u(x)\geq 0, \quad x\in R^N, \end{eqnarray} for sufficiently large $\lambda$, $2 < p < \frac{2N}{N-1}$ for $N \geq 2$. $V(x)$ is a real continuous function on $R^N$. Using variational methods we prove the existence of least energy solution $u(x)$ which localize near the potential well int$(V^{-1}(0))$ for $\lambda$ large. Moreover, if the zero sets int$(V^{-1}(0))$ of $V(x)$ include more than one isolated components, then $u_\lambda(x)$ will be trapped around all the isolated components. However, in Laplacian case, when the parameter $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrary small in other components of int$(V^{-1}(0))$. This is the essential difference with the Laplacian problems since the operator $(-\Delta)^{1/2}$ is nonlocal.

Keywords: half Laplacian, Nonlinear Schrödinger equation, least energy solution, variational methods..

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

Citation: Miaomiao Niu, Zhongwei Tang. Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1215-1231. doi: 10.3934/cpaa.2016.15.1215

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]	D. Applebaum, Lévy processes--from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
[3]	T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384. doi: 10.1007/s000330050003.
[4]	J.L. Bona and Y.A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. doi: 10.1016/S0021-7824(97)89957-6.
[5]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.
[6]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Part. Diff. Equa., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[7]	M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507. doi: 10.1063/1.3701574.
[8]	S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh., 128 (1998), 1249-1260. doi: 10.1017/S030821050002730X.
[9]	J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Diff. Equa., 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.
[10]	A. de Bouard and J. C. Saut, Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal., 28 (1997), 1064-1085. doi: 10.1137/S0036141096297662.
[11]	M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.
[12]	M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire., 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7.
[13]	S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216.
[14]	P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142A (2012), 1237-1262. doi: 10.1017/S0308210511000746.
[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]	R.L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}^{N}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.
[17]	R.L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure. Appl. Math., to appear.
[18]	T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.
[19]	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.
[20]	M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlinear Anal., 51 (2002), 1073-1085. doi: 10.1016/S0362-546X(01)00880-X.
[21]	Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.
[22]	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.
[23]	J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975.
[24]	Z. Tang, On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Comput. Math. Appl., 54 (2007), 627-637. doi: 10.1016/j.camwa.2006.12.031.

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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]	D. Applebaum, Lévy processes--from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
[3]	T. Bartsch and Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384. doi: 10.1007/s000330050003.
[4]	J.L. Bona and Y.A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. doi: 10.1016/S0021-7824(97)89957-6.
[5]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.
[6]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. in Part. Diff. Equa., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[7]	M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507. doi: 10.1063/1.3701574.
[8]	S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh., 128 (1998), 1249-1260. doi: 10.1017/S030821050002730X.
[9]	J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Diff. Equa., 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.
[10]	A. de Bouard and J. C. Saut, Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves, SIAM J. Math. Anal., 28 (1997), 1064-1085. doi: 10.1137/S0036141096297662.
[11]	M. del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265. doi: 10.1006/jfan.1996.3085.
[12]	M. del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire., 15 (1998), 127-149. doi: 10.1016/S0294-1449(97)89296-7.
[13]	S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216.
[14]	P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142A (2012), 1237-1262. doi: 10.1017/S0308210511000746.
[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]	R.L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}^{N}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.
[17]	R.L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure. Appl. Math., to appear.
[18]	T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.
[19]	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.
[20]	M. Maris, On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation, Nonlinear Anal., 51 (2002), 1073-1085. doi: 10.1016/S0362-546X(01)00880-X.
[21]	Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.
[22]	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.
[23]	J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983. doi: 10.3934/dcds.2011.31.975.
[24]	Z. Tang, On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Comput. Math. Appl., 54 (2007), 627-637. doi: 10.1016/j.camwa.2006.12.031.

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