On a Generalized Gagliardo–Nirenberg Inequality with Radial Symmetry and Decaying Potentials
Abstract
:1. Introduction
2. Preliminaries
- 1.
- ;
- 2.
- for every , there exists compact such that ;
- 3.
- for every compact .
3. Main Results
4. Embedding in Function Spaces: Continuity
5. Embedding in Function Spaces: Compactness
5.1. Compactness: Higher Regularity
5.2. Compactness: Unified Approach
6. Gagliardo–Nirenberg Inequalities
7. Minimization Problems
8. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Gidas, B.; Ni, W.M.; Nirenberg, L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 1979, 68, 209–243. [Google Scholar] [CrossRef]
- Badiale, M.; Rolando, S. A note on nonlinear elliptic problems with singular potentials. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2006, 17, 1–13. [Google Scholar] [CrossRef]
- Badiale, M.; Guida, M.; Rolando, S. Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: Compactness. Calc. Var. Partial. Differ. Equ. 2015, 2015, 1061–1090. [Google Scholar] [CrossRef]
- Badiale, M.; Guida, M.; Rolando, S. Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence. NoDEA Nonlinear Differ. Equ. Appl. 2016, 23, 67. [Google Scholar] [CrossRef]
- Su, J.; Wang, Z.Q.; Willem, M. Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differ. Equ. 2007, 238, 201–219. [Google Scholar] [CrossRef]
- Su, J.; Wang, Z.; Willem, M. Nonlinear Schrödinger equations with unbounded and decaying potentials. Commun. Contemp. Math. 2007, 9, 571–583. [Google Scholar] [CrossRef]
- Bonheure, D.; Mercuri, C. Embedding theorems and existence results for non-linear Schrödinger-Poisson systems with unbounded and vanishing potentials. J. Differ. Equ. 2011, 251, 1056–1085. [Google Scholar] [CrossRef]
- Alves, C.; Figueiredo, G.; Siciliano, G. Ground state solutions for fractional scalar field equations under a general critical nonlinearity. Commun. Pure Appl. Anal. 2019, 18, 2199–2215. [Google Scholar] [CrossRef]
- Feng, Z.; Su, Y. Lions-type theorem of the fractional Laplacian and applications. Dyn. Partial Differ. Equ. 2021, 18, 211–230. [Google Scholar] [CrossRef]
- Feng, Z.; Su, Y. Ground state solution to the biharmonic equation. Z. Angew. Math. Phys. 2022, 73, 15. [Google Scholar] [CrossRef]
- Felmer, P.; Quaas, A.; Tan, J. Positive solutions of Nonlinear Schrödinger equations with the Fractional Laplacian. Proc. R. Soc. Edinb. Sect. Math. 2012, 142, 1237–1262. [Google Scholar] [CrossRef]
- Felmer, P.; Wang, Y. Radial symmetry of positive solutions involving the fractional Laplacian. Commun. Contemp. Math. 2014, 16, 1350023. [Google Scholar] [CrossRef]
- Frank, R.L.; Lenzmann, E.; Silvestre, L. Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 2016, 69, 1671–1726. [Google Scholar] [CrossRef]
- Nápoli, P.L.D. Symmetry breaking for an elliptic equation involving the fractional Laplacian. Differ. Integral Equ. 2018, 31, 75–94. [Google Scholar] [CrossRef]
- Li, J.; Ma, L. Extremals to new Gagliardo-Nirenberg inequality and ground states. Appl. Math. Lett. 2021, 120, 107266. [Google Scholar] [CrossRef]
- Su, Y.; Feng, Z. Fractional Sobolev embedding with radial potential. J. Differ. Equ. 2022, 340, 1–44. [Google Scholar] [CrossRef]
- Berestycki, H.; Lions, P.L. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 1983, 82, 313–345. [Google Scholar] [CrossRef]
- Ambrosio, V. Nonlinear Fractional Schrödinger Equations in RN; Frontiers in Elliptic and Parabolic Problems; Birkhäuser: Cham, Switzreland, 2021. [Google Scholar]
- Nezza, E.D.; Palatucci, G.; Valdinoci, E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 2012, 136, 521–573. [Google Scholar] [CrossRef]
- Bellazzini, J.; Ghimenti, M.; Mercuri, C.; Moroz, V.; Schaftingen, J.V. Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces. Trans. Am. Math. Soc. 2018, 370, 8285–8310. [Google Scholar] [CrossRef]
- Palatucci, G.; Pisante, A. Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 2014, 50, 799–829. [Google Scholar] [CrossRef]
- Willem, M. Functional Analysis, Fundamentals and Applications; Cornerstones; Birkhäuser: New York, NY, USA, 2013; pp. 1–213. [Google Scholar]
- Rubin, B.S. One-dimensional representation, inversion and certain properties of Riesz potentials of radial functions. Mat. Zametki 1983, 34, 521–533. (In Russian) [Google Scholar] [CrossRef]
- Gulisashvili, A.; Kon, M.A. Exact smoothing properties of Schroödinger semigroups. Amer. J. Math. 1996, 118, 1215–1248. [Google Scholar]
- Lieb, E.H. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. 1983, 18, 349–374. [Google Scholar] [CrossRef]
- Grafakos, L. Classical Fourier Analysis, 2nd ed.; Graduate Texts in Mathematics, 249; Springer: New York, NY, USA, 2008; pp. xvi + 489. [Google Scholar]
- Moroz, V.; Schaftingen, J.V. Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 2015, 367, 6557–6579. [Google Scholar] [CrossRef]
- Sintzoff, P. Symmetry of solutions of a semilinear elliptic equation with unbounded coefficients. Differ. Integral Equ. 2003, 6, 769–786. [Google Scholar] [CrossRef]
- Stein, E.M.; Weiss, G. Introduction to Fourier Analysis on Euclidean Spaces; Princeton University Press: Princeton, NJ, USA, 1971. [Google Scholar]
- Farah, L.G. Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation. J. Evol. Equ. 2016, 16, 193–208. [Google Scholar] [CrossRef]
- Murphy, J. A simple proof of scattering for the intercritical inhomogeneous NLS. Proc. Am. Math. Soc. 2022, 150, 1177–1186. [Google Scholar] [CrossRef]
- Dinh, V.D. Global dynamics for a class of inhomogeneous nonlinear Schrödinger equation with potential. Math. Nachrichten 2021, 294, 672–716. [Google Scholar] [CrossRef]
- Peng, C.; Zhao, D. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discret. Contin. Dyn. Syst. B 2019, 24, 3335–3356. [Google Scholar] [CrossRef]
- Cuccagna, S.; Tarulli, M. On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential. J. Math. Anal. Appl. 2016, 436, 1332–1368. [Google Scholar] [CrossRef]
- Tarulli, M. H2-scattering for systems of weakly coupled fourth-order NLS Equations in low space dimensions. Potential Anal. 2019, 51, 291–313. [Google Scholar] [CrossRef]
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Tarulli, M.; Venkov, G. On a Generalized Gagliardo–Nirenberg Inequality with Radial Symmetry and Decaying Potentials. Mathematics 2024, 12, 8. https://doi.org/10.3390/math12010008
Tarulli M, Venkov G. On a Generalized Gagliardo–Nirenberg Inequality with Radial Symmetry and Decaying Potentials. Mathematics. 2024; 12(1):8. https://doi.org/10.3390/math12010008
Chicago/Turabian StyleTarulli, Mirko, and George Venkov. 2024. "On a Generalized Gagliardo–Nirenberg Inequality with Radial Symmetry and Decaying Potentials" Mathematics 12, no. 1: 8. https://doi.org/10.3390/math12010008