# American Institute of Mathematical Sciences

January  2020, 19(1): 103-112. doi: 10.3934/cpaa.2020006

## Remarks on singular trudinger-moser type inequalities

 School of Mathematics, Renmin University of China, Beijing 100872, China

Received  August 2018 Revised  April 2019 Published  July 2019

Fund Project: The work is supported by the National Science Foundation of China (Grant No. 11401575)

Let
 $\Omega\subset\mathbb{R}^n$
be a bounded domain. Let
 $F: \mathbb{R}^n\rightarrow[0, +\infty)$
be a convex function of class
 $C^2(\mathbb{R}^n\setminus\{0\})$
, which is even and positively homogeneous of degree
 $1$
. For such a function
 $F$
, there exist two positive constants
 $a_1\leq a_2$
such that
 $a_1|\xi|\leq F(\xi)\leq a_2|\xi|\; (\forall\xi\in\mathbb{R}^n)$
. Therefore,
 $(\int_\Omega F(\nabla u)^n dx)^{1/n}$
and
 $(\int_{\mathbb{R}^n}(F(\nabla u)^n+\tau |u|^n)dx)^{1/n}$
 $(\tau>0)$
are equivalent with the standard norms on
 $W^{1, n}_0(\Omega)$
and
 $W^{1, n}(\mathbb{R}^n)$
respectively. In this paper, we prove that
 \begin{align*} \sup\limits_{u\in W^{1, n}_0(\Omega), \int_\Omega F(\nabla u)^n dx\leq1}\int_\Omega \frac{e^{\lambda|u|^{\frac{n}{n-1}}}}{F^0(x)^{\beta}}dx<+\infty \Leftrightarrow\frac{\lambda}{\lambda_n}+\frac{\beta}{n}\leq1 \end{align*}
and
 \begin{align*} \sup\limits_{u\in W^{1, n}(\mathbb{R}^n), \int_{\mathbb{R}^n}(F(\nabla u)^n+\tau |u|^n)dx\leq1}\int_{\mathbb{R}^n}\frac{e^{\lambda|u|^{\frac{n}{n-1}}}-\sum_{k = 0}^{n-2}\frac{\lambda^k|u|^{\frac{nk}{n-1}}}{k!}}{F^0(x)^{\beta}}dx<\infty\nonumber\ \\ \Leftrightarrow\frac{\lambda}{\lambda_n}+\frac{\beta}{n}\leq1, \end{align*}
where
 $F^0$
is the polar function of
 $F$
,
 $\lambda>0$
,
 $\beta\in[0, n)$
,
 $\tau>0$
,
 $\lambda_n = n^{\frac{n}{n-1}}\kappa_n^{\frac{1}{n-1}}$
and
 $\kappa_n$
is the volume of the unit Wulff ball. Extremal functions for above two supremums are also considered.
Citation: Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006
##### References:
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Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var., 54 (2015), 2341-2366. doi: 10.1007/s00526-015-0867-5. Google Scholar [7] J. do Ó, $N$-Laplacian equations in $\mathbb{R}^N$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315. doi: 10.1155/S1085337597000419. Google Scholar [8] M. Flucher, Extremal functions for Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514. Google Scholar [9] S. Iula and G. Mancini, Extremal functions for singular Moser-Trudinger embeddings, Nonlinear Analysis, 156 (2017), 215-248. doi: 10.1016/j.na.2017.02.029. Google Scholar [10] M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in $\mathbb{R}^N$, Math. Ann., 351 (2011), 781-804. doi: 10.1007/s00208-010-0618-z. Google Scholar [11] X. M. Li and Y. Y. Yang, Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space, J. Differential Equations, 264 (2018), 4901-4943. doi: 10.1016/j.jde.2017.12.028. Google Scholar [12] Y. X. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192. Google Scholar [13] Y. X. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648. doi: 10.1360/04ys0050. Google Scholar [14] Y. X. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^N$, Ind. Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137. Google Scholar [15] K.-C. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3. Google Scholar [16] G. Z. Lu and Y. Y. Yang, The sharp constant and extremal functions for Trudinger-Moser inequalities involving $L^p$ norms, Discrete and Continuous Dynamical Systems, 25 (2009), 963-979. doi: 10.3934/dcds.2009.25.963. Google Scholar [17] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. doi: 10.1512/iumj.1971.20.20101. Google Scholar [18] R. Panda, Nontrivial solution of a quasilinear elliptic equation with critical growth in $\mathbb{R}^n$, Proc. Indian Acad. Sci. (Math. Sci.), 105 (1995), 425-444. doi: 10.1007/BF02836879. Google Scholar [19] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013. Google Scholar [20] M. Struwe, Critical points of embeddings of $H_0^{1,n}$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 425–464. Google Scholar [21] N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. doi: 10.1512/iumj.1968.17.17028. Google Scholar [22] G. F. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320. doi: 10.1016/j.aim.2011.12.001. Google Scholar [23] G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations, 252 (2012), 1668-1700. doi: 10.1016/j.jde.2011.08.001. Google Scholar [24] A. F. Yuan and Z. Y. Huang, An improved singular Trudinger-Moser inequality in dimension two, Turkish J. Math., 40 (2016), 874-883. doi: 10.3906/mat-1501-63. Google Scholar [25] A. F. Yuan and X. B. Zhu, An improved singular Trudinger-Moser inequality in unit ball, J. Math. Anal. Appl., 435 (2016), 244-252. doi: 10.1016/j.jmaa.2015.10.038. Google Scholar [26] Y. Y. Yang, A sharp form of Moser-Trudinger inequality in high dimension, J. Funct. Anal., 239 (2006), 100-126. doi: 10.1016/j.jfa.2006.06.002. Google Scholar [27] Y. Y. Yang, A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface, Trans. Amer. Math. Soc., 359 (2007), 5761-5776. doi: 10.1090/S0002-9947-07-04272-9. Google Scholar [28] Y. Y. Yang, Trudinger-Moser inequalities on complete noncompact Riemannian manifolds, J. Funct. Anal., 263 (2012), 1894-1938. doi: 10.1016/j.jfa.2012.06.019. Google Scholar [29] Y. Y. Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal., 262 (2012), 1679-1704. doi: 10.1016/j.jfa.2011.11.018. Google Scholar [30] Y. Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differential Equations, 258 (2015), 3161-3193. doi: 10.1016/j.jde.2015.01.004. Google Scholar [31] Y. Y. Yang, A Trudinger-Moser inequality on compact Riemannian surface involving Gaussian curvature, J. Geom. Anal., 26 (2016), 2893-2913. doi: 10.1007/s12220-015-9653-z. Google Scholar [32] Y. Y. Yang and X. B. Zhu, An improved Hardy-Trudinger-Moser inequality, Ann. Global Anal. Geom., 49 (2016), 23-41. doi: 10.1007/s10455-015-9478-9. Google Scholar [33] Y. Y. Yang and X. B. Zhu, Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two, J. Funct. Anal., 272 (2017), 3347-3374. doi: 10.1016/j.jfa.2016.12.028. Google Scholar [34] C. L. Zhou and C. Q. Zhou, Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian, Commun. Pure Appl. Anal., 17 (2018), 2309-2328. doi: 10.3934/cpaa.2018110. Google Scholar [35] C. L. Zhou and C. Q. Zhou, Moser-Trudinger inequality involving the anisotropic Dirichlet norm $(\int_{ \Omega}F^N(\nabla u)dx)^{1/N}$ on $W_0^{1,N}( \Omega)$, J. Funct. Anal., 276 (2019), 2901-2935. doi: 10.1016/j.jfa.2018.12.001. Google Scholar [36] J. Y. Zhu, Improved Moser-Trudinger inequality involving $L^p$-norm in ndimensions, Adv. Nonlinear Study, 14 (2014), 273-293. doi: 10.1515/ans-2014-0202. Google Scholar [37] X. B. Zhu, A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularity, Sci. China Math., 62 (2019), 699-718. doi: 10.1007/s11425-017-9174-2. Google Scholar

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##### References:
 [1] Ad imurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations Appl., 13 (2007), 585-603. doi: 10.1007/s00030-006-4025-9. Google Scholar [2] Y. Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, 13 (2010), 2394-2426. Google Scholar [3] A. Alvino, V. Ferone, G. Trombetti and P.-L. Lions, Convex symmetrization and applications, Ann. Inst. Henri Poincaré, 14 (1997), 275-293. doi: 10.1016/S0294-1449(97)80147-3. Google Scholar [4] L. Carleson and A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math., 110 (1986), 113–127. Google Scholar [5] D. M. Cao, Nontrivial solution of semilinear elliptic equations with critical exponent in $\mathbb{R}^2$, Commun. Partial Differential Equations, 17 (1992), 407-435. doi: 10.1080/03605309208820848. Google Scholar [6] G. Csato and P. Roy, Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions, Calc. Var., 54 (2015), 2341-2366. doi: 10.1007/s00526-015-0867-5. Google Scholar [7] J. do Ó, $N$-Laplacian equations in $\mathbb{R}^N$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315. doi: 10.1155/S1085337597000419. Google Scholar [8] M. Flucher, Extremal functions for Trudinger-Moser inequality in 2 dimensions, Comment. Math. Helv., 67 (1992), 471-497. doi: 10.1007/BF02566514. Google Scholar [9] S. Iula and G. Mancini, Extremal functions for singular Moser-Trudinger embeddings, Nonlinear Analysis, 156 (2017), 215-248. doi: 10.1016/j.na.2017.02.029. Google Scholar [10] M. Ishiwata, Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in $\mathbb{R}^N$, Math. Ann., 351 (2011), 781-804. doi: 10.1007/s00208-010-0618-z. Google Scholar [11] X. M. Li and Y. Y. Yang, Extremal functions for singular Trudinger-Moser inequalities in the entire Euclidean space, J. Differential Equations, 264 (2018), 4901-4943. doi: 10.1016/j.jde.2017.12.028. Google Scholar [12] Y. X. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations, 14 (2001), 163-192. Google Scholar [13] Y. X. Li, Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds, Sci. China Ser. A, 48 (2005), 618-648. doi: 10.1360/04ys0050. Google Scholar [14] Y. X. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^N$, Ind. Univ. Math. J., 57 (2008), 451-480. doi: 10.1512/iumj.2008.57.3137. Google Scholar [15] K.-C. Lin, Extremal functions for Moser's inequality, Trans. Amer. Math. Soc., 348 (1996), 2663-2671. doi: 10.1090/S0002-9947-96-01541-3. Google Scholar [16] G. Z. Lu and Y. Y. Yang, The sharp constant and extremal functions for Trudinger-Moser inequalities involving $L^p$ norms, Discrete and Continuous Dynamical Systems, 25 (2009), 963-979. doi: 10.3934/dcds.2009.25.963. Google Scholar [17] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1971), 1077-1092. doi: 10.1512/iumj.1971.20.20101. Google Scholar [18] R. Panda, Nontrivial solution of a quasilinear elliptic equation with critical growth in $\mathbb{R}^n$, Proc. Indian Acad. Sci. (Math. Sci.), 105 (1995), 425-444. doi: 10.1007/BF02836879. Google Scholar [19] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367. doi: 10.1016/j.jfa.2004.06.013. Google Scholar [20] M. Struwe, Critical points of embeddings of $H_0^{1,n}$ into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 425–464. Google Scholar [21] N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. doi: 10.1512/iumj.1968.17.17028. Google Scholar [22] G. F. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Adv. Math., 230 (2012), 294-320. doi: 10.1016/j.aim.2011.12.001. Google Scholar [23] G. F. Wang and C. Xia, Blow-up analysis of a Finsler-Liouville equation in two dimensions, J. Differential Equations, 252 (2012), 1668-1700. doi: 10.1016/j.jde.2011.08.001. Google Scholar [24] A. F. Yuan and Z. Y. Huang, An improved singular Trudinger-Moser inequality in dimension two, Turkish J. Math., 40 (2016), 874-883. doi: 10.3906/mat-1501-63. Google Scholar [25] A. F. Yuan and X. B. Zhu, An improved singular Trudinger-Moser inequality in unit ball, J. Math. Anal. Appl., 435 (2016), 244-252. doi: 10.1016/j.jmaa.2015.10.038. Google Scholar [26] Y. Y. Yang, A sharp form of Moser-Trudinger inequality in high dimension, J. Funct. Anal., 239 (2006), 100-126. doi: 10.1016/j.jfa.2006.06.002. Google Scholar [27] Y. Y. Yang, A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface, Trans. Amer. Math. Soc., 359 (2007), 5761-5776. doi: 10.1090/S0002-9947-07-04272-9. Google Scholar [28] Y. Y. Yang, Trudinger-Moser inequalities on complete noncompact Riemannian manifolds, J. Funct. Anal., 263 (2012), 1894-1938. doi: 10.1016/j.jfa.2012.06.019. Google Scholar [29] Y. Y. Yang, Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space, J. Funct. Anal., 262 (2012), 1679-1704. doi: 10.1016/j.jfa.2011.11.018. Google Scholar [30] Y. Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differential Equations, 258 (2015), 3161-3193. doi: 10.1016/j.jde.2015.01.004. Google Scholar [31] Y. Y. Yang, A Trudinger-Moser inequality on compact Riemannian surface involving Gaussian curvature, J. Geom. Anal., 26 (2016), 2893-2913. doi: 10.1007/s12220-015-9653-z. Google Scholar [32] Y. Y. Yang and X. B. Zhu, An improved Hardy-Trudinger-Moser inequality, Ann. Global Anal. Geom., 49 (2016), 23-41. doi: 10.1007/s10455-015-9478-9. Google Scholar [33] Y. Y. Yang and X. B. Zhu, Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two, J. Funct. Anal., 272 (2017), 3347-3374. doi: 10.1016/j.jfa.2016.12.028. Google Scholar [34] C. L. Zhou and C. Q. Zhou, Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian, Commun. Pure Appl. Anal., 17 (2018), 2309-2328. doi: 10.3934/cpaa.2018110. Google Scholar [35] C. L. Zhou and C. Q. Zhou, Moser-Trudinger inequality involving the anisotropic Dirichlet norm $(\int_{ \Omega}F^N(\nabla u)dx)^{1/N}$ on $W_0^{1,N}( \Omega)$, J. Funct. Anal., 276 (2019), 2901-2935. doi: 10.1016/j.jfa.2018.12.001. Google Scholar [36] J. Y. Zhu, Improved Moser-Trudinger inequality involving $L^p$-norm in ndimensions, Adv. Nonlinear Study, 14 (2014), 273-293. doi: 10.1515/ans-2014-0202. Google Scholar [37] X. B. Zhu, A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularity, Sci. China Math., 62 (2019), 699-718. doi: 10.1007/s11425-017-9174-2. Google Scholar
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