A weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space

Francisco S. B. Albuquerque, Sami Aouaoui

DOI: http://dx.doi.org/10.12775/TMNA.2019.027


We establish a version of the Trudinger-Moser inequality involving unbounded or decaying radial weights in weighted Sobolev spaces. In the light of this inequality and using a minimax procedure we also study existence of solutions for a class of quasilinear elliptic problems involving exponential critical growth.


Trudinger-Moser inequality; quasilinear elliptic problems; weight functions; Exponential critical growth; Mountain pass theorem

Full Text:



S. Adachi and K. Tanaka, Trudinger type inequalities in RN and their best exponents, Proc. Amer. Math. Soc. 128 (2000), 2051–2057.

R. Adams, Sobolev Spaces, Academic Press, New York, 1975.

Adimurthi, Existence of Positive solutions of the semilinear Dirichlet problem with critical growth for the N -Laplacian, Ann. Sci. Norm. Super. Pisa 17 (1990), 393–413.

Adimurthi and K. Sandeep, A singular Moser–Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl. 13 (2007), 585–603.

Adimurthi, and Y. Yang, An interpolation of Hardy inequality and Trudinger–Moser inequality in RN and its applications, Int. Math. Res. Not. IMRN, vol. 2010, no. 13, 2394–2426.

F.S.B. Albuquerque, Sharp constant and extremal function for weighted Trudinger-Moser type inequalities in R2 , J. Math. Anal. Appl. 421 (2015), 963–970.

F.S.B. Albuquerque, Standing wave solutions for a class of nonhomogeneous systems in dimension two, Complex Var. Elliptic Equ. 61 (2016), 1157–1175.

F.S.B. Albuquerque, C.O. Alves and E.S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in R2 , J. Math. Anal. Appl. 409 (2014), 1021–1031.

C.O. Alves, Multiplicity of solutions for a class of elliptic problem in R2 with Neumann conditions, J. Differential Equations 219 (2005), 20–39.

C.O. Alves, J.M.B. do Ó and O.H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth, Nonlinear Anal. 56 (2004), 781–791.

A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (2005), 117–144.

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

S. Aouaoui, On some semilinear elliptic equation involving exponential growth, Appl. Math. Lett. 33 (2014), 23–28.

S. Aouaoui, Multiplicity result for some degenerate elliptic problem involving critical exponential growth in RN , Differential Integral Equations 28 (2015), no. 3–4, 255–270.

S. Aouaoui, Three nontrivial solutions for some elliptic equation involving the N -Laplacian, Electron. J. Qual. Theory Differ. Equ. 2 (2015), 1–12.

S. Aouaoui, On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth, Comm. Pure Appl. Anal. 16 (2017), 1767–1784.

S. Aouaoui, On some quasilinear equation with critical exponential growth at infinity and a singular behavior at the origin, J. Elliptic Parabol. Equ. 4 (2018), 27–50.

D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2 , Comm. Partial Differential Equations 17 (1992), 407–435.

D.G. de Figueiredo, O.H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), 139–153.

D.G. de Figueiredo, J.M. do Ó and B. Ruf, On an inequality by N. Trudinger and J. Moser and related elliptic equations, Comm. Pure Appl. Math. 55 (2002), 1–18.

J.M.B. do Ó, Quasilinear elliptic equations with exponential nonlinearities, Comm. Appl. Nonlin. Anal. 2 (1995), 63–72.

J.M.B. do Ó, Semilinear Dirichlet problems for the n-Laplacian in Rn with nonlinearities in critical growth range, Differential Integral Equations 9 (1996), 967–979.

J.M.B. do Ó, N -Laplacian equations in RN with critical growth, Abstr. Appl. Anal. 2 (1997), 301–315.

J.M.B. do Ó and M. de Souza, On a class of singular Trudinger–Moser inequalities, Math. Nachr. 284 (2011), 1754–1776.

J.M.B. do Ó, M. de Souza, E. de Medeiros and U. Severo, An improvement for the Trudinger–Moser inequality and applications, J. Differential Equations 256 (2014), 1317–1349.

J.M.B. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl. 345 (2008), 286–304.

J.M.B. do Ó and B. Ruf, On a Schrödinger equation with periodic potential and critical growth in R2 , NoDEA Nonlinear Differential Equations Appl. 13 (2006), 167–192.

J.M.B do Ó, F. Sani and J. Zhang, Stationary nonlinear Schrödinger equations in R2 with potential vanishing at infinity, Ann. Mat. Pura Appl. 196 (2017), 363–393.

S. Kesavan, Symmetrization and Applications, Series in Analysis, vol. 3, World Scientific, 2006.

Y. Li and B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in Rn , Indiana Univ. Math. J. 57 (2008), 451–480.

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092.

G. Polya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, 1951.

B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in R2 , J. Funct. Anal. 219 (2005), no. 2, 340–367.

P. Sintzoff and M. Willem, A semilinear elliptic equation on RN with unbounded coefficients, Variational and Topological Methods in the Study of Nonlinear Phenomena (Pisa, 2000), Progr. Nonlinear Differential Equations Appl., vol. 49, Birkhäuser Boston, Boston, MA, 2002, 105–113.

W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162.

J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Comm. Contemp. Math. 9 (2007), 571–583.

J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations 238 (2007), 20-1-219.

N.S. Trudinger, On the embedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–484.

Y. Yang, Existence of positive solutions to quasilinear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal. 262 (2012), 1679–1704.

Y. Yang, Trudinger–Moser inequalities on complete noncompact Riemannian manifolds, J. Funct. Anal. 263 (2012), 1894–1938.


  • There are currently no refbacks.

Partnerzy platformy czasopism