Abstract and Applied Analysis

Abstract and Applied Analysis / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 957468 | https://doi.org/10.1155/2013/957468

Baiyu Liu, "A Liouville Type Result for Schrödinger Equation on Half-Spaces", Abstract and Applied Analysis, vol. 2013, Article ID 957468, 4 pages, 2013. https://doi.org/10.1155/2013/957468

A Liouville Type Result for Schrödinger Equation on Half-Spaces

Academic Editor: Svatoslav Staněk
Received10 Sep 2013
Accepted20 Nov 2013
Published28 Nov 2013

Abstract

We consider a nonlinear Schrödinger equation with a singular potential on half spaces. Using a Hardy-type inequality and the moving plane method, we obtain a Liouville type result for its nonnegative solutions.

1. Introduction

Recently, properties of nontrivial solutions for nonlinear elliptic equations on half spaces have attracted a great deal of attention from physicians and mathematicians; see, for example, [15].

In this paper, we consider nonnegative solutions of the following Schrödinger equation with a singular potential on the half-space: where,,, and

Equation (1) is related to the Grushin type equation with critical exponent and the Webster scalar curvature equation [6, 7].

We are interested in the Liouville type result for nonnegative solutions of (1). This work is motivated by some monotonicity results and Liouville type results for elliptic equations on half-spaces; see, for example, [2, 3]. In [2], Dancer found some sufficient conditions for nonlinear termsuch that the positive bounded solutionofwith Dirichlet boundary value condition is monotone increasing in. Guo [3] considered nonnegative solutions for the elliptic system, and obtained some sufficient conditions for and , under which system (3) admits only trivial solution.

Letbe the space given by the completion ofunder the norm. We say thatis a weak solution of (1) ifsatisfies for all.

Using a Hardy-type inequality and the moving plane method in integral forms [810], we obtain the following Liouville type result.

Theorem 1. Letbe a nonnegative weak solution of (1) with. Then,.

Remark 2. For a weak solution, by using a regularity lifting method [8], we know that, for all bounded smooth domain. Hence, it is a classical solution.

2. Preliminary

In this section, we prepare some lemmas.

Firstly, we recall the Hardy-Sobolev inequality in the half space; see [1113].

Lemma 3. Let; then,

This inequality plays a crucial role in estimating the singular potential term in the following proof.

In the following, we assume that is a nonnegative weak solution of (1) with. We are going to use the method of moving plane in the half-space.

For each, let For, we writewhich is the reflected point ofwith respect to the hyperplaneand define

Then, direct computation gives here.

For, we have,,, and. Therefore,

Defineand. Clearly,, and. Define

The heart of our argument is the following lemma.

Lemma 4. There exists a, such that, for, if, then

Proof. For, let be defined by
Testing (9) inwith function, we obtain The left hand side of (13) is Hence, we derive where Using Lemma 3, we have
For,,, which implies
By using Hölder inequality, we verify that
Putting (17), (19), and (20) into (15) and using the assumption, we then deduce that
Moreover, by the Sobolev inequality, we know that
Combine the above inequality with (21) to get
Now we claim that Notice that, for,. Hence, Since,. Thus (24) is valid.
Now, lettingin (23), by using dominated convergence theorem, we obtain If .
One can choose, whereis the best constant in the Sobolev inequality.

Using Lemma 4, we now can start the moving plane process as the following Lemma.

Lemma 5. There is a, such that, for all,

Proof. Since, using Sobolev inequality, we have. Choosesmall enough such that whereis the same as in Lemma 4.
Hence, for all, which is a contradiction to Lemma 4, if . That is to say, which implies that , for.

Now we move the hyperplane upwards by increasing the value ofcontinuously as long as (27) holds. We will show that the hyperplane will be moved to the infinity. Precisely, define By the result of Lemma 5,.

Lemma 6. We have .

Proof. Suppose .
On one hand, by continuity we know that, for all, which means
On the other hand, by the definition of, there is that satisfy (i), as, and (ii) , for all. By Lemma 4, we get . By using the dominated convergence theorem, we obtain which is a contradiction to (32).

3. Proof of Theorem 1

In this section, we prove Theorem 1.

Sinceis a superharmonic continuous function in(see Remark 2), we have eitherinorin.

Ifin, then there is somesatisfying. Moreover, by continuity, there is a, such that, for all. By using Lemma 6, we know thatis increasing with respect toin. Thus,for alland. Hence, which contradicts the fact that.

Therefore,in.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (no. 11201025) and the Fundamental Research Funds for the Central Universities (FRF-TP-12-106A).

References

  1. L. Caffarelli and L. Silvestre, “An extension problem related to the fractional Laplacian,” Communications in Partial Differential Equations, vol. 32, no. 7–9, pp. 1245–1260, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. E. N. Dancer, “Some notes on the method of moving planes,” Bulletin of the Australian Mathematical Society, vol. 46, no. 3, pp. 425–434, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. Y. Guo, “Non-existence, monotonicity for positive solutions of semilinear elliptic system in +N,” Communications in Contemporary Mathematics, vol. 12, no. 3, pp. 351–372, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. B. Liu and L. Ma, “Symmetry results for elliptic Schrödinger systems on half spaces,” Journal of Mathematical Analysis and Applications, vol. 401, no. 1, pp. 259–268, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. B. Liu and L. Ma, “Symmetry results for decay solutions of semilinear elliptic systems on half spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 6, pp. 3167–3177, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. D. Castorina, I. Fabbri, G. Mancini, and K. Sandeep, “Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations,” Journal of Differential Equations, vol. 246, no. 3, pp. 1187–1206, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. S. Chen and L. Lin, “Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients,” Science China, vol. 54, no. 2, pp. 221–242, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, vol. 4, American Institute of Mathematical Sciences (AIMS), 2010. View at: MathSciNet
  9. Y. Guo and J. Liu, “Liouville type theorems for positive solutions of elliptic system in N,” Communications in Partial Differential Equations, vol. 33, no. 1–3, pp. 263–284, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. C. Li and L. Ma, “Uniqueness of positive bound states to Schrödinger systems with critical exponents,” SIAM Journal on Mathematical Analysis, vol. 40, no. 3, pp. 1049–1057, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. S. Chen and S. Li, “Hardy-Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 2, pp. 324–348, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. J. Tidblom, “A Hardy inequality in the half-space,” Journal of Functional Analysis, vol. 221, no. 2, pp. 482–495, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. V. G. Maz'ja, Sobolev Spaces, Springer, Berlin, germany, 1985. View at: MathSciNet

Copyright © 2013 Baiyu Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

536 Views | 440 Downloads | 0 Citations
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.