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    In memory of professor Rouhuai Wang (1924-2001): A pioneering Chinese researcher in partial differential equations
February  2016, 36(2): 577-600. doi: 10.3934/dcds.2016.36.577

Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875

2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15261, United States

3. 

Department of Mathematics, Wayne State University, Detroit, MI 48202, United States

Received  August 2014 Published  August 2015

In this paper, we apply the sharp Adams-type inequalities for the Sobolev space $W^{m,\frac{n}{m}}\left( \mathbb{R} ^{n}\right) $ for any positive real number $m$ less than $n$, established by Ruf and Sani [46] and Lam and Lu [30,31], to study polyharmonic equations in $\mathbb{R}^{2m}$. We will consider the polyharmonic equations in $\mathbb{R}^{2m}$ of the form \[ \left( I-\Delta\right) ^{m}u=f(x,u)\text{ in }% \mathbb{R} ^{2m}. \] We study the existence of the nontrivial solutions when the nonlinear terms have the critical exponential growth in the sense of Adams' inequalities on the entire Euclidean space. Our approach is variational methods such as the Mountain Pass Theorem ([5]) without Palais-Smale condition combining with a version of a result due to Lions ([39,40]) for the critical growth case. Moreover, using the regularity lifting by contracting operators and regularity lifting by combinations of contracting and shrinking operators developed in [14] and [11], we will prove that our solutions are uniformly bounded and Lipschitz continuous. Finally, using the moving plane method of Gidas, Ni and Nirenberg [22,23] in integral form developed by Chen, Li and Ou [12] together with the Hardy-Littlewood-Sobolev type inequality instead of the maximum principle, we prove our positive solutions are radially symmetric and monotone decreasing about some point. This appears to be the first work concerning existence of nontrivial nonnegative solutions of the Bessel type polyharmonic equation with exponential growth of the nonlinearity in the whole Euclidean space.
Citation: Jiguang Bao, Nguyen Lam, Guozhen Lu. Polyharmonic equations with critical exponential growth in the whole space $ \mathbb{R}^{n}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 577-600. doi: 10.3934/dcds.2016.36.577
References:
[1]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives,, Ann. of Math. (2) 128 (1988), 128 (1988), 385.  doi: 10.2307/1971445.  Google Scholar

[2]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n-$Laplacian,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 17 (1990), 393.   Google Scholar

[3]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, Nonlinear perturbations of a periodic elliptic problem with critical growth,, J. Math. Anal. Appl., 260 (2001), 133.  doi: 10.1006/jmaa.2001.7442.  Google Scholar

[4]

C. O. Alves, J. M. do Ó and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $ \mathbbR ^{2}$ involving critical growth,, Nonlinear Anal., 56 (2004), 781.  doi: 10.1016/j.na.2003.06.003.  Google Scholar

[5]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[6]

F. V. Atkinson and L. A. Peletier, Ground states and Dirichlet problems for $-\Delta u=f(u)$ in $ \mathbbR ^{2}$,, Arch. Rational Mech. Anal., 96 (1986), 147.  doi: 10.1007/BF00251409.  Google Scholar

[7]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[8]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $ \mathbbR ^{2}$,, Comm. Partial Differential Equations, 17 (1992), 407.  doi: 10.1080/03605309208820848.  Google Scholar

[9]

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser,, Bull. Sci. Math. (2), 110 (1986), 113.   Google Scholar

[10]

K. C. Chang, Methods in Nonlinear Analysis,, Springer Monographs in Mathematics, (2005).   Google Scholar

[11]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Series on Differential Equations & Dynamical Systems, (2010).   Google Scholar

[12]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[13]

W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems,, J. Math. Anal. Appl., 377 (2011), 744.  doi: 10.1016/j.jmaa.2010.11.035.  Google Scholar

[14]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adv. Math., 226 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[15]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $ \mathbbR ^n$,, Comm. Pure Appl. Math., 45 (1992), 1217.  doi: 10.1002/cpa.3160451002.  Google Scholar

[16]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $ \mathbbR ^{2}$ with nonlinearities in the critical growth range,, Calc. Var. Partial Differential Equations, 3 (1995), 139.  doi: 10.1007/BF01205003.  Google Scholar

[17]

J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $ \mathbbR ^N$,, J. Differential Equations, 246 (2009), 1363.  doi: 10.1016/j.jde.2008.11.020.  Google Scholar

[18]

M. J. Esteban, Nonsymmetric ground state of symmetric variational problems,, Comm. Pure Appl. Math., 44 (1991), 259.  doi: 10.1002/cpa.3160440205.  Google Scholar

[19]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions,, Comment. Math. Helv., 67 (1992), 471.  doi: 10.1007/BF02566514.  Google Scholar

[20]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds,, Comm. Math. Helv., 68 (1993), 415.  doi: 10.1007/BF02565828.  Google Scholar

[21]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains,, Lecture Notes in Mathematics, (1991).  doi: 10.1007/978-3-642-12245-3.  Google Scholar

[22]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[23]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $ \mathbbR ^N$,, in Mathematical analysis and applications, 7a (1981), 369.   Google Scholar

[24]

X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potential,, Communications on Pure and Applied Analysis, 10 (2011), 1111.  doi: 10.3934/cpaa.2011.10.1111.  Google Scholar

[25]

X. Han and G. Lu, On regularity of solutions to an integral system associated with Bessel potentials,, International Journal of Mathematics, 23 (2012).  doi: 10.1142/S0129167X12500516.  Google Scholar

[26]

O. Kavian, Introduction à la Théorie des Points Critiques et Applications Aux Problèmes Elliptiques,, Springer-Verlag, (1993).   Google Scholar

[27]

H. Kozono, T. Sato and H. Wadade, Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality,, Indiana Univ. Math. J., 55 (2006), 1951.  doi: 10.1512/iumj.2006.55.2743.  Google Scholar

[28]

N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $n-$Laplacian type with critical exponential growth in $R^n$,, J. Funct. Anal., 262 (2012), 1132.  doi: 10.1016/j.jfa.2011.10.012.  Google Scholar

[29]

N. Lam and G. Lu, Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth,, Discrete Contin. Dyn. Syst., 32 (2012), 2187.  doi: 10.3934/dcds.2012.32.2187.  Google Scholar

[30]

N. Lam and G. Lu, Sharp Adams type inequalities in Sobolev spaces $W^{m,\fracnm}(R^n)$ for arbitrary integer $m$,, J. Differential Equations, 253 (2012), 1143.  doi: 10.1016/j.jde.2012.04.025.  Google Scholar

[31]

N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement-free argument,, J. Differential Equations, 255 (2013), 298.  doi: 10.1016/j.jde.2013.04.005.  Google Scholar

[32]

N. Lam and G. Lu, The Moser-Trudinger and Adams inequalities and elliptic and subelliptic equations with nonlinearity of exponential growth,, Recent developments in geometry and analysis, 23 (2012), 179.   Google Scholar

[33]

N. Lam and G. Lu, Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications,, Adv. Math., 231 (2012), 3259.  doi: 10.1016/j.aim.2012.09.004.  Google Scholar

[34]

N. Lam, G. Lu and H. Tang, Sharp subcritical Moser-Trudinger inequalities on Heisenberg groups and subelliptic PDEs,, Nonlinear Anal., 95 (2014), 77.  doi: 10.1016/j.na.2013.08.031.  Google Scholar

[35]

Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two,, J. Partial Differential Equations, 14 (2001), 163.   Google Scholar

[36]

Y. Li, Remarks on the extremal functions for the Moser-Trudinger inequality,, Acta Math. Sin. (Engl. Ser.), 22 (2006), 545.  doi: 10.1007/s10114-005-0568-7.  Google Scholar

[37]

Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbbR ^n$,, Indiana Univ. Math. J., 57 (2008), 451.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[38]

K. Lin, Extremal functions for Moser's inequality,, Trans. Amer. Math. Soc., 348 (1996), 2663.  doi: 10.1090/S0002-9947-96-01541-3.  Google Scholar

[39]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I,, Rev. Mat. Iberoamericana, 1 (1985), 145.  doi: 10.4171/RMI/6.  Google Scholar

[40]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II,, Rev. Mat. Iberoamericana, 1 (1985), 45.  doi: 10.4171/RMI/12.  Google Scholar

[41]

P. Montecchiari, Multiplicity results for a class of semilinear elliptic equations on $ \mathbbR ^n$,, Rend. Sem. Mat. Univ. Padova, 95 (1996), 217.   Google Scholar

[42]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.   Google Scholar

[43]

E. S. Noussair, C. A. Swanson and J. Yang, Quasilinear elliptic problem with critical exponents,, Nonlinear Anal. TMA, 20 (1993), 285.  doi: 10.1016/0362-546X(93)90164-N.  Google Scholar

[44]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$,, (Russian) Dokl. Akad. Nauk SSSR, 165 (1965), 36.   Google Scholar

[45]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbbR ^{2}$,, J. Funct. Anal., 219 (2005), 340.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[46]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $ \mathbbR ^n$,, Trans. Amer. Math. Soc., 365 (2013), 645.  doi: 10.1090/S0002-9947-2012-05561-9.  Google Scholar

[47]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).   Google Scholar

[48]

C. Tarsi, Adams' inequality and limiting Sobolev embeddings into Zygmund spaces,, Potential Anal., 37 (2012), 353.  doi: 10.1007/s11118-011-9259-4.  Google Scholar

[49]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.   Google Scholar

[50]

V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations,, (Russian) Dokl. Akad. Nauk SSSR, 138 (1961), 805.   Google Scholar

show all references

References:
[1]

D. R. Adams, A sharp inequality of J. Moser for higher order derivatives,, Ann. of Math. (2) 128 (1988), 128 (1988), 385.  doi: 10.2307/1971445.  Google Scholar

[2]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n-$Laplacian,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 17 (1990), 393.   Google Scholar

[3]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, Nonlinear perturbations of a periodic elliptic problem with critical growth,, J. Math. Anal. Appl., 260 (2001), 133.  doi: 10.1006/jmaa.2001.7442.  Google Scholar

[4]

C. O. Alves, J. M. do Ó and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $ \mathbbR ^{2}$ involving critical growth,, Nonlinear Anal., 56 (2004), 781.  doi: 10.1016/j.na.2003.06.003.  Google Scholar

[5]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[6]

F. V. Atkinson and L. A. Peletier, Ground states and Dirichlet problems for $-\Delta u=f(u)$ in $ \mathbbR ^{2}$,, Arch. Rational Mech. Anal., 96 (1986), 147.  doi: 10.1007/BF00251409.  Google Scholar

[7]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[8]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $ \mathbbR ^{2}$,, Comm. Partial Differential Equations, 17 (1992), 407.  doi: 10.1080/03605309208820848.  Google Scholar

[9]

L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser,, Bull. Sci. Math. (2), 110 (1986), 113.   Google Scholar

[10]

K. C. Chang, Methods in Nonlinear Analysis,, Springer Monographs in Mathematics, (2005).   Google Scholar

[11]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Series on Differential Equations & Dynamical Systems, (2010).   Google Scholar

[12]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[13]

W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems,, J. Math. Anal. Appl., 377 (2011), 744.  doi: 10.1016/j.jmaa.2010.11.035.  Google Scholar

[14]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, Adv. Math., 226 (2011), 2676.  doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[15]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $ \mathbbR ^n$,, Comm. Pure Appl. Math., 45 (1992), 1217.  doi: 10.1002/cpa.3160451002.  Google Scholar

[16]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $ \mathbbR ^{2}$ with nonlinearities in the critical growth range,, Calc. Var. Partial Differential Equations, 3 (1995), 139.  doi: 10.1007/BF01205003.  Google Scholar

[17]

J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $ \mathbbR ^N$,, J. Differential Equations, 246 (2009), 1363.  doi: 10.1016/j.jde.2008.11.020.  Google Scholar

[18]

M. J. Esteban, Nonsymmetric ground state of symmetric variational problems,, Comm. Pure Appl. Math., 44 (1991), 259.  doi: 10.1002/cpa.3160440205.  Google Scholar

[19]

M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions,, Comment. Math. Helv., 67 (1992), 471.  doi: 10.1007/BF02566514.  Google Scholar

[20]

L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds,, Comm. Math. Helv., 68 (1993), 415.  doi: 10.1007/BF02565828.  Google Scholar

[21]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains,, Lecture Notes in Mathematics, (1991).  doi: 10.1007/978-3-642-12245-3.  Google Scholar

[22]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[23]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $ \mathbbR ^N$,, in Mathematical analysis and applications, 7a (1981), 369.   Google Scholar

[24]

X. Han and G. Lu, Regularity of solutions to an integral equation associated with Bessel potential,, Communications on Pure and Applied Analysis, 10 (2011), 1111.  doi: 10.3934/cpaa.2011.10.1111.  Google Scholar

[25]

X. Han and G. Lu, On regularity of solutions to an integral system associated with Bessel potentials,, International Journal of Mathematics, 23 (2012).  doi: 10.1142/S0129167X12500516.  Google Scholar

[26]

O. Kavian, Introduction à la Théorie des Points Critiques et Applications Aux Problèmes Elliptiques,, Springer-Verlag, (1993).   Google Scholar

[27]

H. Kozono, T. Sato and H. Wadade, Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality,, Indiana Univ. Math. J., 55 (2006), 1951.  doi: 10.1512/iumj.2006.55.2743.  Google Scholar

[28]

N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $n-$Laplacian type with critical exponential growth in $R^n$,, J. Funct. Anal., 262 (2012), 1132.  doi: 10.1016/j.jfa.2011.10.012.  Google Scholar

[29]

N. Lam and G. Lu, Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth,, Discrete Contin. Dyn. Syst., 32 (2012), 2187.  doi: 10.3934/dcds.2012.32.2187.  Google Scholar

[30]

N. Lam and G. Lu, Sharp Adams type inequalities in Sobolev spaces $W^{m,\fracnm}(R^n)$ for arbitrary integer $m$,, J. Differential Equations, 253 (2012), 1143.  doi: 10.1016/j.jde.2012.04.025.  Google Scholar

[31]

N. Lam and G. Lu, A new approach to sharp Moser-Trudinger and Adams type inequalities: A rearrangement-free argument,, J. Differential Equations, 255 (2013), 298.  doi: 10.1016/j.jde.2013.04.005.  Google Scholar

[32]

N. Lam and G. Lu, The Moser-Trudinger and Adams inequalities and elliptic and subelliptic equations with nonlinearity of exponential growth,, Recent developments in geometry and analysis, 23 (2012), 179.   Google Scholar

[33]

N. Lam and G. Lu, Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications,, Adv. Math., 231 (2012), 3259.  doi: 10.1016/j.aim.2012.09.004.  Google Scholar

[34]

N. Lam, G. Lu and H. Tang, Sharp subcritical Moser-Trudinger inequalities on Heisenberg groups and subelliptic PDEs,, Nonlinear Anal., 95 (2014), 77.  doi: 10.1016/j.na.2013.08.031.  Google Scholar

[35]

Y. Li, Moser-Trudinger inequality on compact Riemannian manifolds of dimension two,, J. Partial Differential Equations, 14 (2001), 163.   Google Scholar

[36]

Y. Li, Remarks on the extremal functions for the Moser-Trudinger inequality,, Acta Math. Sin. (Engl. Ser.), 22 (2006), 545.  doi: 10.1007/s10114-005-0568-7.  Google Scholar

[37]

Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbbR ^n$,, Indiana Univ. Math. J., 57 (2008), 451.  doi: 10.1512/iumj.2008.57.3137.  Google Scholar

[38]

K. Lin, Extremal functions for Moser's inequality,, Trans. Amer. Math. Soc., 348 (1996), 2663.  doi: 10.1090/S0002-9947-96-01541-3.  Google Scholar

[39]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I,, Rev. Mat. Iberoamericana, 1 (1985), 145.  doi: 10.4171/RMI/6.  Google Scholar

[40]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II,, Rev. Mat. Iberoamericana, 1 (1985), 45.  doi: 10.4171/RMI/12.  Google Scholar

[41]

P. Montecchiari, Multiplicity results for a class of semilinear elliptic equations on $ \mathbbR ^n$,, Rend. Sem. Mat. Univ. Padova, 95 (1996), 217.   Google Scholar

[42]

J. Moser, A sharp form of an inequality by N. Trudinger,, Indiana Univ. Math. J., 20 (): 1077.   Google Scholar

[43]

E. S. Noussair, C. A. Swanson and J. Yang, Quasilinear elliptic problem with critical exponents,, Nonlinear Anal. TMA, 20 (1993), 285.  doi: 10.1016/0362-546X(93)90164-N.  Google Scholar

[44]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$,, (Russian) Dokl. Akad. Nauk SSSR, 165 (1965), 36.   Google Scholar

[45]

B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $ \mathbbR ^{2}$,, J. Funct. Anal., 219 (2005), 340.  doi: 10.1016/j.jfa.2004.06.013.  Google Scholar

[46]

B. Ruf and F. Sani, Sharp Adams-type inequalities in $ \mathbbR ^n$,, Trans. Amer. Math. Soc., 365 (2013), 645.  doi: 10.1090/S0002-9947-2012-05561-9.  Google Scholar

[47]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).   Google Scholar

[48]

C. Tarsi, Adams' inequality and limiting Sobolev embeddings into Zygmund spaces,, Potential Anal., 37 (2012), 353.  doi: 10.1007/s11118-011-9259-4.  Google Scholar

[49]

N. S. Trudinger, On imbeddings into Orlicz spaces and some applications,, J. Math. Mech., 17 (1967), 473.   Google Scholar

[50]

V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations,, (Russian) Dokl. Akad. Nauk SSSR, 138 (1961), 805.   Google Scholar

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