Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities

By D. Smets. 2005

207 citations recorded by the source corpus.

We study a time-independent nonlinear Schrodinger equation with an attractive inverse square potential and a nonautonomous nonlinearity whose power is the critical Sobolev exponent. The problem shares a strong resemblance with the prescribed scalar curvature problem on the standard sphere. Particular attention is paid to the blow-up possibilities, i.e. the critical points at infinity of the corresponding variational problem. Due to the strong singularity in the potential, some new phenomenon appear. A complete existence result is obtained in dimension 4 using a detailed analysis of the gradient flow lines.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/411303b0d763392f3650e20f38d5a1278e0475eb

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In Applied Mathematics

1. Shuangjie Peng
2. Shusen Yan
3. Daomin Cao
4. Zhitao Zhang
5. Thomas Bartsch
6. M. Fall
7. R. Musina
8. Veronica Felli

In Mathematical Physics

1. Shuangjie Peng
2. Daomin Cao
3. Shusen Yan
4. Zhitao Zhang
5. Thomas Bartsch
6. M. Fall
7. R. Musina
8. Veronica Felli

Citations in the corpus

• Elliptic Equations with Multi-Singular Inverse-Square Potentials and Critical Nonlinearity (2006, background): “Besides the already mentioned papers [14, 15, 19, 27], we quote, among others, [1, 9, 10, 12, 25, 26 ].”
• Hardy—Poincaré inequalities with boundary singularities (2010, background): “1 includes the nonexistence result proved by Smets [13], Proposition 2.”
• Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential (2010, background): “For other results on the global compactness, see [ 24 ,25].”
• Existence and non-existence of solutions to elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities (2007, background): “Moreover, some interesting results such as the existence and non-existence of the extremal functions and the symmetry and non-symmetry of the extremal functions were discussed; see [3, 9, 16, 28, 30, 34, 35], for instance.”
• Hardy–Sobolev extremals, hyperbolic symmetry and scalar curvature equations (2009, unknown): “, in [8] and [26], we will look for critical points coming from changes in the topology of sublevels.”