# Weighted Trudinger - Moser Inequalities and Applications

M. Calanchi, B. RufTrudinger - Moser inequalities provide continuous embeddings in the borderline cases of the standard Sobolev embeddings, in which the embeddings into Lebesgue L^p spaces break down. One is led to consider their natural generalization, which are embeddings into Orlicz spaces with corresponding maximal growth functions which are of exponential type. Some parameters come up in the description of these growth functions. The parameter ranges for which embeddings exist increase by the use of weights in the Sobolev norm, and one is led to consider weighted TM inequalities. Some interesting cases are presented for special weights in dimension two, with applications to mean field equations of Liouville type.Full text

- Keywords
- Trudinger - Moser inequalities; Orlicz spaces; maximal growth functions; weighted TM inequalities.
- References
- 1. Yudovich V.I. Some Estimates Connected with Integral Operators and with Solutions of Elliptic Equations. Dokl. Akad. Nauk SSSR, 1961, vol. 138, pp. 805-808.

2. Peetre J. Espaces d' interpolation et theoreme de Soboleff. Ann. Inst. Fourier, 1966, vol. 16, pp. 279-317. DOI: 10.5802/aif.232

3. Pohozaev S.I. The Sobolev Embedding in the Case pl = n. Proc. of the Technical Scientific Conference on Advances of Scientific Research, 1964 - 1965, Mathematics Section, (Moskov. Energet. Inst., Moscow), pp. 158-170.

4. Trudinger N.S. On Imbeddings into Orlicz Spaces and Some Applications. Journal of Applied Mathematics and Mechanics, 1967, vol. 17, pp. 473-483. DOI: 10.1512/iumj.1968.17.17028

5. Moser J. Moser J. A Sharp Form of an Inequality by N. Trudinger. Indiana University Mathematics Journal, 1970/71, vol. 20, pp. 1077-1092. DOI: 10.1512/iumj.1971.20.20101

6. Liouville J., Sur l' equation aux derivees partielles. Journal de Mathematiques Pures et Appliquees, 1853, vol. 18, pp. 71-72.

7. Caglioti E., Lions P.L., Marchioro C., Pulvirenti M. A Special Class of Stationary Flows for Two-Dimensional Euler Equations: a Statistical Mechanics Description. Communications in Mathematical Physics, 1992, vol. 143, no. 3, pp. 501-525. DOI: 10.1007/BF02099262

8. Caglioti E., Lions P.L., Marchioro C., Pulvirenti M. A Special Class of Stationary Flows for Two-Dimensional Euler Equations: a Statistical Mechanics Description. II. Communications in Mathematical Physics, 1995, vol. 174, no. 2, pp. 229-260. DOI: 10.1007/BF02099602

9. Chanillo S., Kiessling M. Rotational Symmetry of Solutions of Some Nonlinear Problems in Statistical Mechanics and in Geometry. Communications in Mathematical Physics, 1994, vol. 160, no. 2, pp. 217-238. DOI: 10.1007/BF02103274

10. Kiessling M.K.-H. Statistical Mechanics of Classical Particles with Logarithmic Interactions. Communications on Pure and Applied Mathematics, 1993, vol. 46, pp. 27-56. DOI: 10.1002/cpa.3160460103

11. Tarantello G. Multiple Condensate Solutions for the Chern - Simons - Higgs Theory. Journal of Mathematical Physics, 1996, vol. 37, pp. 3769-3796. DOI: 10.1063/1.531601

12. Tarantello G. Analytical Aspects of Liouville-Type Equations with Singular Sources. Handbook of Differential Equations (M. Chipot and P. Quittner, eds.), Elsevier, North Holland, 2004, pp. 491-592.

13. Li Y.Y. Harnack Type Inequality: the Method of Moving Planes. Communications in Mathematical Physics, 1999, vol. 200, pp. 421-444. DOI: 10.1007/s002200050536

14. Chen C.C., Lin C.S. Mean Field Equations of Liouville Type with Singular Data: Sharper Estimates. Discrete and Continuous Dynamic Systems, 2010, vol. 28, no. 3, pp. 123-127.

15. Calanchi M., Terraneo E. Non-radial Maximizers for Functionals with Exponential Nonlinearity in R^2. Advanced Nonlinear Studies, 2005, vol. 5, pp. 337-350.

16. Adimurthi, Sandeep K. A Singular Moser - Trudinger Embedding and Its Applications. Nonlinear Differential Equations and Applications, 2007, vol. 13, issue 5-6, pp. 585-603. DOI: 10.1007/s00030-006-4025-9

17. Calanchi M., Ruf B. On a Trudinger - Moser Type Inequality with Logarithmic Weights. Journal of Differential Equations, 2015, no. 3, pp. 258-263. DOI: 10.1016/j.jde.2014.11.019

18. Calanchi M. Some Weighted Inequalitie of Trudinger - Moser Type in Progress. Nonlinear Differential Equations and Applications, Birkhauser, 2014, vol. 85, pp. 163-174.

19. Calanchi M., Ruf B. Trudinger - Moser Type Inequalities with Logarithmic Weights in Dimension N. Nonlinear Anal., 2015, vol. 121, pp. 403-411.

20. Kufner A. Weighted Sobolev Spaces. John Wiley & Sons Ltd, 1985.

21. Leckband M.A. An Integral Inequality with Applications. Transactions of the American Mathematical Society, 1984, vol. 283, no. 1, pp. 157-168. DOI: 10.1090/S0002-9947-1984-0735413-7

22. de Figueiredo G., Miyagaki O.H., Ruf B. Elliptic Equations in R^2 with Nonlinearities in the Critical Growth Range. Calc. Var. Partial Differential Equations, 1995, vol. 3, no. 2, pp. 139-153. DOI: 10.1007/BF01205003