## International Journal of Differential Equations

### Ground State for the Schrödinger Operator with the Weighted Hardy Potential

#### Abstract

We establish the existence of ground states on ${\Bbb R}^{N}$ for the Laplace operator involving the Hardy-type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.

#### Article information

Source
Int. J. Differ. Equ., Volume 2011 (2011), Article ID 358087, 26 pages.

Dates
Accepted: 17 July 2011
First available in Project Euclid: 25 January 2017

https://projecteuclid.org/euclid.ijde/1485313232

Digital Object Identifier
doi:10.1155/2011/358087

Zentralblatt MATH identifier
1234.35157

#### Citation

Chabrowski, J.; Tintarev, K. Ground State for the Schrödinger Operator with the Weighted Hardy Potential. Int. J. Differ. Equ. 2011 (2011), Article ID 358087, 26 pages. doi:10.1155/2011/358087. https://projecteuclid.org/euclid.ijde/1485313232

#### References

• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2nd edition, 1983.
• M. Murata, “Structure of positive solutions to $(-\Delta +V)u=0$ in ${\mathbb{R}}^{N}$,” Duke Mathematical Journal, vol. 53, no. 4, pp. 869–943, 1986.
• Y. Pinchover and K. Tintarev, “A ground state alternative for singular Schrödinger operators,” Journal of Functional Analysis, vol. 230, no. 1, pp. 65–77, 2006.
• Y. Pinchover and K. Tintarev, “Ground state alternative for p-Laplacian with potential term,” Calculus of Variations and Partial Differential Equations, vol. 28, no. 2, pp. 179–201, 2007.
• V. G. Maz'ja, Sobolev Spaces, Springer Series in Soviet Mathematics, Springer, Berlin, Germany, 1985.
• W. Allegretto, “Principal eigenvalues for indefinite-weight elliptic problems in ${\mathbb{R}}^{N}$,” Proceedings of the American Mathematical Society, vol. 116, no. 3, pp. 701–706, 1992.
• K. J. Brown, C. Cosner, and J. Fleckinger, “Principal eigenvalues for problems with indefinite weight function on ${\mathbb{R}}^{N}$,” Proceedings of the American Mathematical Society, vol. 109, no. 1, pp. 147–155, 1990.
• K. J. Brown and A. Tertikas, “The existence of principal eigenvalues for problems with indefinite weight function on IR,” Proceedings of the Royal Society of Edinburgh A, vol. 123, no. 3, pp. 561–569, 1993.
• Z. Jin, “Principal eigenvalues with indefinite weight functions,” Transactions of the American Mathematical Society, vol. 349, no. 5, pp. 1945–1959, 1997.
• L. Leadi and A. Yechoui, “Principal eigenvalue in an unbounded domain with indefinite potential,” Nonlinear Differential Equations and Applications, vol. 17, no. 4, pp. 391–409, 2010.
• N. B. Rhouma, “Principal eigenvalues for indefinite weight problems in all of ${\mathbb{R}}^{N}$,” Proceedings of the American Mathematical Society, vol. 131, no. 12, pp. 3747–3755, 2004.
• D. Smets, “A concentration–-compactness lemma with applications to singular eigenvalue problems,” Journal of Functional Analysis, vol. 167, no. 2, pp. 463–480, 1999.
• A. Szulkin and M. Willem, “Eigenvalue problems with indefinite weight,” Studia Mathematica, vol. 135, no. 2, pp. 191–201, 1999.
• T. V. Anoop, M. Lucia, and M. Ramaswamy, “Eigenvalue problems with weights in Lorentz spaces,” Calculus of Variations and Partial Differential Equations, vol. 36, no. 3, pp. 355–376, 2009.
• N. Visciglia, “A note about the generalized Hardy-Sobolev inequality with potential in ${L}^{p,d}({\mathbb{R}}^{N})$,” Calculus of Variations and Partial Differential Equations, vol. 24, no. 2, pp. 167–184, 2005.
• J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren der Mathematischen Wissenschaften, no. 22, Springer, Berlin, Germany, 1976.
• R. A. Hunt, “On L(p, q) spaces,” L'Enseignement Mathématique, vol. 12, no. 2, pp. 249–276, 1966.
• G. G. Lorentz, “Some new functional spaces,” Annals of Mathematics, vol. 51, pp. 37–55, 1950.
• A. Tertikas, “Critical phenomena in linear elliptic problems,” Journal of Functional Analysis, vol. 154, no. 1, pp. 42–66, 1998.
• Y. Pinchover, A. Tertikas, and K. Tintarev, “A Liouville-type theorem for the p-Laplacian with potential term,” Annales de l'Institut Henri Poincaré C, vol. 25, no. 2, pp. 357–368, 2008.
• J. Chabrowski, “On the nonlinear Neumann problem involving the critical Sobolev exponent and Hardy potential,” Revista Matemática Complutense, vol. 17, no. 1, pp. 195–227, 2004.
• J. Chen, “Exact local behavior of positive solutions for a semilinear elliptic equation with Hardy term,” Proceedings of the American Mathematical Society, vol. 132, no. 11, pp. 3225–3229, 2004.
• P. Han, “Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential,” Proceedings of the American Mathematical Society, vol. 135, no. 2, pp. 365–372, 2007.
• L. Caffarelli, R. Kohn, and L. Nirenberg, “First order interpolation inequalities with weights,” Compositio Mathematica, vol. 53, no. 3, pp. 259–275, 1984.