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On a weighted Adachi-Tanaka type Trudinger-Moser inequality in nonradial Sobolev spaces. (English) Zbl 1467.35012

Summary: In this article we establish a Trudinger-Moser inequality of Adachi-Tanaka type in nonradial weighted Sobolev spaces for functions defined on the whole \(\mathbb{R}^N\). The main tools are the Besecovitch covering lemma and a Trudinger-Moser inequality on the whole space established by S. Adachi and K. Tanaka.

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
26D10 Inequalities involving derivatives and differential and integral operators
26D15 Inequalities for sums, series and integrals
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[1] Adachi, S. and Tanaka, K., Trudinger type inequalities inRNand their best exponents.Proc. Amer. Math. Soc.128 (1999), 2051 - 2057. · Zbl 0980.46020
[2] Adams, D. R., A sharp inequality of J. Moser for higher order derivatives.Ann. Math.128 (1988), 385 - 398. · Zbl 0672.31008
[3] Adimurthi and Yang, Y., An interpolation of Hardy inequality and Trudinger- Moser inequality inRNand its applications.Int. Math. Res. Not. IMRN2010 (2010)(13), 2394 - 2426. · Zbl 1198.35012
[4] Albuquerque, F. S. B., Alves, C. O. and Medeiros, E. S., Nonlinear Schr¨odinger equation with unbounded or decaying radial potentials involving exponential critical growth inR2.J. Math. Anal. Appl.409 (2014), 1021 - 1031. · Zbl 1309.35123
[5] Albuquerque, F. S. B. and Aouaoui, S., A weighted Trudinger-Moser type inequality and its applications to quasilinear elliptic problems with critical growth in the whole Euclidean space.Topol. Methods Nonlinear Anal.54 (2019)(1), 109 - 130. · Zbl 1429.35008
[6] Ambrosetti, A., Felli, V. and Malchiodi, A., Ground states of nonlinear Schr¨odinger equations with potentials vanishing at infinity.J. Eur. Math. Soc. 7 (2005), 117 - 144. · Zbl 1064.35175
[7] Calanchi, M. and Ruf, B., Trudinger-Moser type inequalities with logarithmic weights in dimensionN.Nonlinear Anal.121 (2015), 403 - 411. · Zbl 1338.46042
[8] Cao, D., Nontrivial solution of semilinear elliptic equations with critical exponent inR2.Comm. Partial Diff. Equ.17 (1992), 407 - 435. · Zbl 0763.35034
[9] Chang, S. Y. A. and Yang, P., The inequality of Moser and Trudinger and applications to conformal geometry.Comm. Pure Appl. Math.56 (2003), 1135 - 1150. · Zbl 1049.53025
[10] Cianchi, A., Moser-Trudinger trace inequalities.Adv. Math.217 (2008), 2005 - 2044. · Zbl 1138.46020
[11] de Figueiredo, D. G., do ´O, J. M. and Ruf, B., Elliptic equations and systems with critical Trudinger-Moser nonlinearities.Discrete Contin. Dyn. Syst.30 (2011), 455 - 476. · Zbl 1222.35084
[12] de Figueiredo, D. G., Miyagaki, O. H. and Ruf, B., Elliptic equations inR2 with nonlinearities in the critical growth range.Calc. Var. Partial Diff. Equ. 3 (1995)(2), 139 - 153. · Zbl 0820.35060
[13] de Guzm´an, M.,Differentiation of Integrals inRn.Lect. Notes Math. 481. Berlin: Springer 1975. · Zbl 0327.26010
[14] de Souza, M.,On a singular elliptic problem involving critical growth inRN, Nonlinear Diff. Equ. Appl.18 (2011), 199 - 215. · Zbl 1216.35071
[15] de Souza, M., On a class of singular Trudinger-Moser type inequalities for unbounded domains inRN.Appl. Math. Lett.25 (2012)(12), 2100 - 2104. · Zbl 1259.46030
[16] de Souza, M. and do ´O, J. M., On a class of singular Trudinger-Moser type inequalities and its applications.Math. Nachr.284 (2011), 1754 - 1776. · Zbl 1229.35098
[17] de Souza, M. and do ´O, J. M., On singular Trudinger-Moser type inequalities for unbounded domains and their best exponents.Potential Anal.38 (2013)(4), 1091 - 1101. · Zbl 1279.46026
[18] de Souza, M., do ´O, J. M. and Silva, T. S., Quasilinear nonhomogeneous Schr¨odinger equation with critical exponential growth inRn.Topol. Methods Nonlinear Anal.45 (2015), 615 - 639. · Zbl 1376.35071
[19] do ´O, J. M., Semilinear Dirichlet problems for theN-Laplacian inRNwith nonlinearities in the critical growth range.Diff. Integral Equ.9 (1996)(5), 967 - 979. · Zbl 0858.35043
[20] do ´O, J. M.,N-Laplacian equations inRNwith critical growth.Abstr. Appl. Anal.2 (1997), 301 - 315. · Zbl 0932.35076
[21] do ´O, J. M., Sani, F. and Zhang, J., Stationary nonlinear Schr¨odinger equations inR2with potentials vanishing at infinity.Ann. Mat. Pura Appl. (4)196 (2017)(1), 363 - 393. · Zbl 1365.35168
[22] Dong, M. and Lu, G., Best constants and existence of maximizers for weighted Trudinger-Moser inequalities.Calc. Var. Partial Diff. Equ.55 (2016)(4), Art. 88, 26 pp. · Zbl 1364.46026
[23] Fontana, L. and Morpurgo, C., Sharp exponential integrability for critical Riesz potentials and fractional Laplacians onRn.Nonlinear Anal.167 (2018), 85 - 122. · Zbl 1392.46034
[24] Furtado, M. F., Medeiros, E. S. and Severo, U. B., A Trudinger-Moser inequality in a weighted Sobolev space and applications.Math. Nachr.287 (2014), 1255 - 1273. · Zbl 1303.35022
[25] Giacomoni, J. and Sreenadh, K., A multiplicity result to a nonhomogeneous elliptic equation in whole spaceR2.Adv. Math. Sci. Appl.15 (2005), 467 - 488. · Zbl 1220.35072
[26] Ishiwata, M., Nakamura, M. and Wadade, H., On the sharp constant for the weighted Trudinger-Moser type inequality of the scaling invariant form.Ann. Inst. H. Poincar´e Anal. Non Lineaire31 (2014)(2), 297 - 314. · Zbl 1311.46034
[27] Lam, N. and Lu, G., A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument.J. Diff. Equ.255 (2013)(3), 298 - 325. · Zbl 1294.46034
[28] Lam, N. and Lu, G.,N-Laplacian equations inRNwith subcritical and critical growth without the Ambrosetti-Rabinowitz condition.Adv. Nonlinear Stud.13 (2013)(2), 289 - 308. · Zbl 1283.35049
[29] Moser, J., A sharp form of an inequality by N. Trudinger.Indiana Univ. Math. J.20 (1971), 1077 - 1092. · Zbl 0213.13001
[30] Ogawa, T., A proof of Trudinger’s inequality and its application to nonlinear Schr¨odinger equations.Nonlinear Anal.14 (1990), 765 - 769. · Zbl 0715.35073
[31] Ozawa, T., On critical cases of Sobolev’s inequalities.J. Funct. Anal.127 (1995), 259 - 269. · Zbl 0846.46025
[32] Panda, R., Nontrivial solution of a quasilinear elliptic equation with critical growth inRN.Proc. Indian Acad. Sci. Math. Sci.105 (1995), 425 - 444. · Zbl 0854.35035
[33] Poho˘zaev, S. I., The Sobolev embedding in the special casepl=n(in Russian). In:Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964-1965.Moscow: Section Math. Moscow Power Inst. 1965, pp. 158 - 170.
[34] Strichartz, R. S., A note on Trudinger’s extension of Sobolev’s inequalities. Indiana Univ. Math. J.21 (1972), 841 - 842. · Zbl 0241.46028
[35] Tian, G. and Zhu, X. H., A nonlinear inequality of Moser-Trudinger type. Calc. Var. Partial Diff. Equ.10 (2000)(4), 349 - 354. · Zbl 0987.32017
[36] Trudinger, N. S., On the embedding into Orlicz spaces and some applications. J. Math. Mech.17 (1967), 473 - 484. · Zbl 0163.36402
[37] Yang, T. and Zhu, X., A new proof of subcritical Trudinger-Moser inequalities on the whole Euclidean space.J. Partial Diff. Equ.26 (2013)(4), 300 - 304. · Zbl 1313.46045
[38] Yudovi˘c, V. I., Some estimates connected with integral operators and with solutions of elliptic equations (in Russian).Dokl. Akad. Nauk SSSR138 (1961), 804 - 808; Engl. transl.:Soviet Math. Dokl.2 (1961), 746 - 749 · Zbl 0144.14501
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