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Minimal potential results for Schrödinger equations with Neumann boundary conditions

  • Julian Edward EMAIL logo , Steve Hudson and Mark Leckband
Published/Copyright: January 28, 2017

Abstract

We consider the boundary value problem -Δpu=V|u|p-2u-C, where uW1,p(D) is assumed to satisfy Neumann boundary conditions, and D is a bounded domain in n. We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for the product of a sharp Sobolev constant and an Lp norm of V. When p=n, Orlicz norms are used. In many cases, these inequalities are best possible. Applications to linear and non-linear eigenvalue problems are also discussed.

MSC 2010: 35B05; 26D20; 35P15

Communicated by Christopher D. Sogge


Acknowledgements

We gratefully acknowledge helpful conversations with Bao Qin Li and Laura De Carli.

References

[1] G. Astarita and G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw–Hill, New York, 1974. Search in Google Scholar

[2] R. Bañuelos and K. Burdzy, On the “hot spots” conjecture of J. Rauch, J. Funct. Anal. 164 (1999), 1–33. 10.1006/jfan.1999.3397Search in Google Scholar

[3] Y. Belaud, Time-vanishing properties of solutions of some degenerate parabolic equations with strong absorption, Adv. Nonlinear Stud. 1 (2001), no. 2, 117–152. 10.1515/ans-2001-0205Search in Google Scholar

[4] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. (2) 106 (1977), 93–102. 10.2307/1971160Search in Google Scholar

[5] L. De Carli, J. Edward, S. Hudson and M. Leckband, Minimal support results for Schrödinger equations, Forum Math. 27 (2015), no. 1, 343–371. 10.1515/forum-2012-0106Search in Google Scholar

[6] L. De Carli and S. Hudson, Geometric remarks on the level curves of harmonic functions, Bull. Lond. Math. Soc. 42 (2010), no. 1, 83–95. 10.1112/blms/bdp099Search in Google Scholar

[7] A. V. Demyanov and A. I. Nazarov, On the existence of extremal functions in Sobolev embedding theorems with critical exponents, St. Petersburg Math. J. 17 (2006), no. 5, 773–796. 10.1090/S1061-0022-06-00929-0Search in Google Scholar

[8] G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Port. Math. (N.S.) 58 (2001), no. 3, 339–378. Search in Google Scholar

[9] J. Edward, S. Hudson and M. Leckband, Existence problems for the p-Laplacian, Forum Math. 27 (2015), no. 2, 1203–1225. 10.1515/forum-2012-0142Search in Google Scholar

[10] G. Ercole, Absolute continuity of the best Sobolev constant, J. Math. Anal. Appl. 404 (2013), no. 2, 420–428. 10.1016/j.jmaa.2013.03.044Search in Google Scholar

[11] L. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 2002. Search in Google Scholar

[12] M. Fraas and Y. Pinchover, Positive Liouville theorems and asymptotic behavior for p-Laplacian type elliptic equations with a Fuchsian potential, Confluentes Math. 3 (2011), no. 2, 291–323. 10.1142/S1793744211000321Search in Google Scholar

[13] R. L. Frank, E. H. Lieb and R. Seiringer, Equivalence of Sobolev inequalities and Lieb–Thirring inequalities, XVIth International Congress on Mathematical Physics (Prague 2009), World Scientific, Hackensack (2010), 523–535. 10.1142/9789814304634_0045Search in Google Scholar

[14] P. Girao and T. Weth, The shape of extremal functions for Poincaré–Sobolev-type inequalities in a ball, J. Funct. Anal. 237 (2006), 194–223. 10.1016/j.jfa.2006.01.001Search in Google Scholar

[15] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser, Basel, 2006. 10.1007/3-7643-7706-2Search in Google Scholar

[16] I. Holopainen, Quasiregular mappings and the p-Laplace operator, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces, Contemp. Math. 338, American Mathematical Society, Providence (2003), 219–239. 10.1090/conm/338/06075Search in Google Scholar

[17] D. Hundertmark, Some bound state problems in quantum mechanics, Spectral Theory and Mathematical Physics. A Festschrift in Honour of Barry Simon’s 60th Birthday, Proc. Sympos. Pure Math. 76, American Mathematical Society, Providence (2007), 463–496. 10.1090/pspum/076.1/2310215Search in Google Scholar

[18] M. Krasnosel’skií and Y. Rutickii, Convex Functions and Orlicz Spaces, P. Noordhoff, Groningen, 1961. Search in Google Scholar

[19] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in N, J. Funct. Anal. 262 (2012), no. 3, 1132–1165. 10.1016/j.jfa.2011.10.012Search in Google Scholar

[20] N. Lam and G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition, J. Geom. Anal. 24 (2014), no. 1, 118–143. 10.1007/s12220-012-9330-4Search in Google Scholar

[21] A. Lê, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006), 1057–1099. 10.1016/j.na.2005.05.056Search in Google Scholar

[22] M. Leckband, Moser’s inequality on the ball Bn for functions with mean value zero, Comm. Pure Appl. Math. 58 (2005), 789–798. 10.1002/cpa.20056Search in Google Scholar

[23] E. H. Lieb, Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. Amer. Math. Soc. 82 (1976), 751–752. 10.1090/S0002-9904-1976-14149-3Search in Google Scholar

[24] R. McOwen, Partial Differential Equations, Methods and Applications, 2nd ed., Prentice Hall, Upper Saddle River, 2003. Search in Google Scholar

[25] G. V. Rozenblum, Distribution of the discrete spectrum of singular differential operators (in Russian), Dokl. Akad. Nauk SSSR 202 (1972), 1012–1015; translatiom in Sov. Math. Dokl. 13 (1972), 245–249. Search in Google Scholar

[26] I. Seo, On minimal support properties of solutions of Schrödinger equations, J. Math. Anal. Appl. 414 (2014), no. 1, 21–28. 10.1016/j.jmaa.2013.12.047Search in Google Scholar

[27] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta. Math. 138 (1977), 219–240. 10.1007/BF02392316Search in Google Scholar

[28] Y. Yang, Moser–Trudinger inequality for functions with mean value zero, Nonlinear Anal. 66 (2007), 2742–2755. 10.1016/j.na.2006.04.004Search in Google Scholar

Received: 2015-5-7
Revised: 2016-9-28
Published Online: 2017-1-28
Published in Print: 2017-11-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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