The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy

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March 2012, 11(2): 587-626. doi: 10.3934/cpaa.2012.11.587

The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy

Robert Magnus ^1, and Olivier Moschetta ^2,

1.	University Science Institute, University of Iceland, Dunhaga 3, 107--Reykjavik, Iceland
2.	Reykjavik Junior College, Lækjargata 7, 101--Reykjavik, Iceland

Received September 2010 Revised January 2011 Published October 2011

Abstract
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The equation $-\varepsilon^2 \Delta u+F(V(x),u)=0$ is considered in $R^n$. It is assumed that $V$ possesses a set of critical points $B$ for which the values of $V$ and $D^2V$ satisfy certain compactness and uniformity properties. Under appropriate conditions on $F$ the problem is shown to possess for each $b\in B$ and small $\varepsilon>0$ a solution that concentrates at $b$ and has detailed uniformity and decay properties. This enables the construction of solutions that concentrate at arbitrary subsets of $B$ as $\varepsilon\to 0$. Examples are given in which $B$ is infinite and $V$ non-periodic.

Keywords: Non-linear Schrödinger equation, multibumps..

Mathematics Subject Classification: Primary: 35J60; Secondary: 35Q51, 35Q5.

Citation: Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure and Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587

References:

[1]	N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Analysis, 234 (2006), 277-320. doi: 10.1016/j.jfa.2005.11.010.
[2]	S. Angenent, The shadowing lemma for elliptic PDE, in "Dynamics of Infinite-dimensional Systems," (eds. S. N. Chow and J. K. Hale), Springer-Verlag, New York, (1987), 7-22.
[3]	V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^{n}$, Comm. Pure. Appl. Math., 45 (1992), 1217-1270. doi: 10.1002/cpa.3160451002.
[4]	A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Analysis, 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.
[5]	C. Gui, Existence of multi-bump solutions for non-linear Schrödinger equations via variational method, Comm. in PDE, 21 (1996), 787-820. doi: 10.1080/03605309608821208.
[6]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{n}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[7]	R. Magnus, The implicit function theorem and multibump solutions of periodic partial differential equations, Proceedings of the Royal Society of Edinburgh, 136A (2006), 559-583. doi: 10.1017/S0308210500005060.
[8]	R. Magnus, A scaling approach to bumps and multi-bumps for nonlinear partial differential equations, Proceedings of the Royal Society of Edinburgh, 136A (2006), 585-614. doi: 10.1017/S0308210500005072.
[9]	R. Magnus and O. Moschetta, Non-degeneracy of perturbed solutions of semilinear partial differential equations, Ann. Aca. Scient. Fennicae, 35 (2010), 75-86. doi: 10.5186/aasfm.2010.3505.
[10]	O. Moschetta, "The Non-linear Schrödinger Equation: Non-degeneracy and Infinite-bump Solutions," Ph.D thesis, University of Iceland, 2010.
[11]	Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Commun. Partial Diff. Eq., 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.
[12]	Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-255. doi: 10.1007/BF02161413.
[13]	J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660. doi: 10.2307/2313800.
[14]	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.
[15]	M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327.
[16]	M. del Pino, P. Felmer and K. Tanaka, An elementary construction of complex patterns in nonlinear Schrödinger equations, Nonlinearity, 15 (2002), 1653-1671.
[17]	P. Rabier and C. Stuart, Exponential decay of the solutions of quasilinear second-order equations and Pohozaev identities, Journal of Differential Equations, 165 (2000), 199-234. doi: 10.1006/jdeq.1999.3749.
[18]	L. Schwartz, "Théorie des distributions," Hermann, Paris, 1966.
[19]	C. Stuart, An introduction to elliptic equations in $\mathbb{R}^{n}$, in "Nonlinear Functional Analysis and Applications to Differential Equations," World Sci. Publ. River Edge, NJ, (1998), 237-285.
[20]	N. Thandi, "On the Existence of Infinite Bump Solutions of Nonlinear Schrödinger Equations with Periodic Potentials," Ph.D thesis, University of Wisconsin-Madison, 1995.
[21]	J-L. Verger-Gaugry, Covering a ball with smaller equal balls in $\mathbb{R}^{N}$, Discrete and Computational Geometry, 33 (2005), 143-155. doi: 10.1007/s00454-004-2916-2.
[22]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.
[23]	M. Weinstein, Modulational stability of the ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 567-576.

show all references

References:

[1]	N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Analysis, 234 (2006), 277-320. doi: 10.1016/j.jfa.2005.11.010.
[2]	S. Angenent, The shadowing lemma for elliptic PDE, in "Dynamics of Infinite-dimensional Systems," (eds. S. N. Chow and J. K. Hale), Springer-Verlag, New York, (1987), 7-22.
[3]	V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^{n}$, Comm. Pure. Appl. Math., 45 (1992), 1217-1270. doi: 10.1002/cpa.3160451002.
[4]	A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Analysis, 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0.
[5]	C. Gui, Existence of multi-bump solutions for non-linear Schrödinger equations via variational method, Comm. in PDE, 21 (1996), 787-820. doi: 10.1080/03605309608821208.
[6]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{n}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[7]	R. Magnus, The implicit function theorem and multibump solutions of periodic partial differential equations, Proceedings of the Royal Society of Edinburgh, 136A (2006), 559-583. doi: 10.1017/S0308210500005060.
[8]	R. Magnus, A scaling approach to bumps and multi-bumps for nonlinear partial differential equations, Proceedings of the Royal Society of Edinburgh, 136A (2006), 585-614. doi: 10.1017/S0308210500005072.
[9]	R. Magnus and O. Moschetta, Non-degeneracy of perturbed solutions of semilinear partial differential equations, Ann. Aca. Scient. Fennicae, 35 (2010), 75-86. doi: 10.5186/aasfm.2010.3505.
[10]	O. Moschetta, "The Non-linear Schrödinger Equation: Non-degeneracy and Infinite-bump Solutions," Ph.D thesis, University of Iceland, 2010.
[11]	Y. G. Oh, Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Commun. Partial Diff. Eq., 13 (1988), 1499-1519. doi: 10.1080/03605308808820585.
[12]	Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-255. doi: 10.1007/BF02161413.
[13]	J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660. doi: 10.2307/2313800.
[14]	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.
[15]	M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327.
[16]	M. del Pino, P. Felmer and K. Tanaka, An elementary construction of complex patterns in nonlinear Schrödinger equations, Nonlinearity, 15 (2002), 1653-1671.
[17]	P. Rabier and C. Stuart, Exponential decay of the solutions of quasilinear second-order equations and Pohozaev identities, Journal of Differential Equations, 165 (2000), 199-234. doi: 10.1006/jdeq.1999.3749.
[18]	L. Schwartz, "Théorie des distributions," Hermann, Paris, 1966.
[19]	C. Stuart, An introduction to elliptic equations in $\mathbb{R}^{n}$, in "Nonlinear Functional Analysis and Applications to Differential Equations," World Sci. Publ. River Edge, NJ, (1998), 237-285.
[20]	N. Thandi, "On the Existence of Infinite Bump Solutions of Nonlinear Schrödinger Equations with Periodic Potentials," Ph.D thesis, University of Wisconsin-Madison, 1995.
[21]	J-L. Verger-Gaugry, Covering a ball with smaller equal balls in $\mathbb{R}^{N}$, Discrete and Computational Geometry, 33 (2005), 143-155. doi: 10.1007/s00454-004-2916-2.
[22]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. doi: 10.1007/BF02096642.
[23]	M. Weinstein, Modulational stability of the ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 567-576.

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