Abstract
In this paper, our goal is to establish two kinds of generalized version of Lions-type theorems for the p-Laplacian and then apply them to investigate the existence of solutions of quasilinear elliptic equations with the critical exponent.
Similar content being viewed by others
Data Availability
Our manuscript has no associated data. No data was used or created in this study.
References
Abdellaoui, B., Peral, I.: Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential. Ann. Mat. Pura Appl. 182(3), 247–270 (2003)
Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53(6), 1629–1663 (2004)
Badiale, M., Rolando, S.: A note on nonlinear elliptic problems with singular potentials. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 17(1), 1–13 (2006)
Badiale, M., Benci, V., Rolando, S.: A nonlinear elliptic equation with singular potential and applications to nonlinear field equations. J. Eur. Math. Soc. 9(3), 355–381 (2007)
Badiale, M., Guida, M., Rolando, S.: Compactness and existence results for the p-Laplace equation. J. Math. Anal. Appl. 451(1), 345–370 (2017)
Baldelli, L., Brizi, Y., Filippucci, R.: On symmetric solutions for (p, q)-Laplacian equations in \(R^N\) with critical terms. J. Geom. Anal. 32(4), 1–25 (2022)
Benci, V., Fortunato, D.: Variational Methods in Nonlinear Field Equations, Solitary Waves, Hylomorphic Solitons and Vortices. Springer Monogr. Math., Springer, Cham (2014)
Brézis, H., Lieb, E.H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)
Buchheit, A.A., Rjasanow, S.: Ground state of the Frenkel-Kontorova model with a globally deformable substrate potential. Phys. D 406, 132298 (2020)
Carrião, P.C., Demarque, R., Miyagaki, O.H.: Nonlinear biharmonic problems with singular potentials. Commun. Pure Appl. Anal. 13(6), 2141–2154 (2014)
Feng, Z., Su, Y.: Lions-type theorem of the fractional Laplacian and applications. Dyn. Partial Differ. Equ. 18(3), 211–230 (2021)
Fife, P.C.: Asymptotic states for equations of reaction and diffusion. Bull. Am. Math. Soc. 84(5), 693–728 (1978)
Filippucci, R., Pucci, P., Robert, F.: On a p-Laplace equation with multiple critical nonlinearities. J. Math. Pures Appl. 91(2), 156–177 (2009)
Ghoussoub, N., Yuan, C.: Multiple solutions for quasi-linear PDEs involving the critical Hardy and Sobolev exponents. Trans. Am. Math. Soc. 352(12), 5703–5743 (2000)
Kawohl, B., Horak, J.: On the geometry of the \(p\)-Laplacian operator. Discret. Contin. Dyn. Syst. Ser. S 10(4), 799–813 (2017)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Math. Iberoam. 1(2), 145–201 (1985)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Math. Iberoam. 1(1), 45–121 (1985)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The Locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(2), 109–145 (1984)
Lions, P.L.: The concentration-compactness principle in the calculus of variations. The Locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(4), 223–283 (1984)
Monticelli, D.D., Punzo, F., Sciunzi, B.: Nonexistence of stable solutions to quasilinear elliptic equations on Riemannian manifolds. J. Geom. Anal. 27(4), 3030–3050 (2017)
Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3–4), 799–829 (2014)
Su, J., Wang, Z., Willem, M.: Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differ. Equ. 238, 201–219 (2007)
Su, J., Tian, R.: Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Commun. Pure Appl. Anal. 9(4), 885–904 (2010)
Su, J., Tian, R.: Weighted Sobolev type embeddings and coercive quasilinear elliptic equations on \({\mathbb{R} }^{N}\). Proc. Am. Math. Soc. 140(3), 891–903 (2012)
Terracini, S.: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differ. Equ. 1(2), 241–264 (1996)
Vincent, C., Phatak, S.: Accurate momentum-space method for scattering by nuclear and Coulomb potentials. Phys. Rev. 10, 391–394 (1974)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing financial interests or other conflict of interest that could have appeared to influence the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by Key Program of University Natural Science Research Fund of Anhui Province (KJ2020A0292).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Feng, Z., Su, Y. Lions-Type Properties for the p-Laplacian and Applications to Quasilinear Elliptic Equations. J Geom Anal 33, 99 (2023). https://doi.org/10.1007/s12220-022-01150-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-022-01150-4