November  2021, 20(11): 3871-3886. doi: 10.3934/cpaa.2021135

Bounds for subcritical best Sobolev constants in W1, p

Dip. di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, Villa Toeplitz via G.B. Vico 46, 21100 Varese, Italy

Received  February 2021 Revised  June 2021 Published  November 2021 Early access  August 2021

This paper aims at establishing fine bounds for subcritical best Sobolev constants of the embeddings
$ W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega),\quad 1\leq q< \begin{cases} \frac{Np}{N-p},& 1\leq p
where
$ N\geq p\geq1 $
and
$ \Omega $
is a bounded smooth domain in
$ \mathbb{R}^{N} $
or the whole space. The Sobolev limiting case
$ p = N $
is also covered by means of a limiting procedure.
Citation: Lele Du. Bounds for subcritical best Sobolev constants in W1, p. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3871-3886. doi: 10.3934/cpaa.2021135
References:
[1]

P. d'Avenia and C. Ji, Semiclassical states for a magnetic nonlinear Schrödinger equation with exponentical critical growth in $\mathbb{R}^{2}$, preprint, arXiv: 2106.05962. doi: 10.1016/j.na.2016.12.004.

[2]

A. Alvino, A limit case of the Sobolev inequality in Lorentz spaces, Rend. Accad. Sci. Fis. Mat. Napoli, 44 (1977), 105-112. 

[3]

C. O. Alves and S. H. M. Souto, Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differential Equations, 234 (2007), 464-484.  doi: 10.1016/j.jde.2006.12.006.

[4]

C. O. AlvesM. A. S. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differ. Equ., 43 (2012), 537-554.  doi: 10.1007/s00526-011-0422-y.

[5]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598. 

[6]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[7]

G. CeramiD. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 341-350. 

[8]

D. CassaniB. Ruf and C. Tarsi, Optimal Sobolev type inequalities in Lorentz spaces, Potential Anal., 39 (2013), 265-285.  doi: 10.1007/s11118-012-9329-2.

[9]

D. CassaniF. Sani and C. Tarsi, Equivalent Moser type inequalities in $\mathbb{R}^{2}$ and the zero mass case, J. Funct. Anal., 267 (2014), 4236-4263.  doi: 10.1016/j.jfa.2014.09.022.

[10]

D. CassaniC. Tarsi and J. Zhang, Bounds for best constants in subcritical Sobolev embeddings, Nonlinear Anal., 187 (2019), 438-449.  doi: 10.1016/j.na.2019.05.012.

[11]

X. Chen and J. Yang, Improved Sobolev inequalities and critical problems, Commun. Pure Appl. Anal., 19 (2020), 3673-3695.  doi: 10.3934/cpaa.2020162.

[12]

V. G. Maz'ya, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl., 1 (1960), 882-885. 

[13]

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.

[14]

J. M. do Ó and S. M. Souto, On a class of nonlinear Schrödinger equations in $\mathbb{R}^{2}$ involving critical growth, J. Differ. Equ., 20 (2001), 289-311.  doi: 10.1006/jdeq.2000.3946.

[15]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. 

[16]

P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.  doi: 10.1512/iumj.1986.35.35036.

[17]

X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763.  doi: 10.2307/2154740.

[18]

X. Ren and J. Wei, Counting peaks of solutions to some quasilinear elliptic equations with large exponents, J. Differ. Equ., 117 (1995), 28-55.  doi: 10.1006/jdeq.1995.1047.

[19]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[20]

G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

show all references

References:
[1]

P. d'Avenia and C. Ji, Semiclassical states for a magnetic nonlinear Schrödinger equation with exponentical critical growth in $\mathbb{R}^{2}$, preprint, arXiv: 2106.05962. doi: 10.1016/j.na.2016.12.004.

[2]

A. Alvino, A limit case of the Sobolev inequality in Lorentz spaces, Rend. Accad. Sci. Fis. Mat. Napoli, 44 (1977), 105-112. 

[3]

C. O. Alves and S. H. M. Souto, Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differential Equations, 234 (2007), 464-484.  doi: 10.1016/j.jde.2006.12.006.

[4]

C. O. AlvesM. A. S. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differ. Equ., 43 (2012), 537-554.  doi: 10.1007/s00526-011-0422-y.

[5]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry, 11 (1976), 573-598. 

[6]

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.

[7]

G. CeramiD. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 341-350. 

[8]

D. CassaniB. Ruf and C. Tarsi, Optimal Sobolev type inequalities in Lorentz spaces, Potential Anal., 39 (2013), 265-285.  doi: 10.1007/s11118-012-9329-2.

[9]

D. CassaniF. Sani and C. Tarsi, Equivalent Moser type inequalities in $\mathbb{R}^{2}$ and the zero mass case, J. Funct. Anal., 267 (2014), 4236-4263.  doi: 10.1016/j.jfa.2014.09.022.

[10]

D. CassaniC. Tarsi and J. Zhang, Bounds for best constants in subcritical Sobolev embeddings, Nonlinear Anal., 187 (2019), 438-449.  doi: 10.1016/j.na.2019.05.012.

[11]

X. Chen and J. Yang, Improved Sobolev inequalities and critical problems, Commun. Pure Appl. Anal., 19 (2020), 3673-3695.  doi: 10.3934/cpaa.2020162.

[12]

V. G. Maz'ya, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl., 1 (1960), 882-885. 

[13]

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.

[14]

J. M. do Ó and S. M. Souto, On a class of nonlinear Schrödinger equations in $\mathbb{R}^{2}$ involving critical growth, J. Differ. Equ., 20 (2001), 289-311.  doi: 10.1006/jdeq.2000.3946.

[15]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. 

[16]

P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.  doi: 10.1512/iumj.1986.35.35036.

[17]

X. Ren and J. Wei, On a two-dimensional elliptic problem with large exponent in nonlinearity, Trans. Amer. Math. Soc., 343 (1994), 749-763.  doi: 10.2307/2154740.

[18]

X. Ren and J. Wei, Counting peaks of solutions to some quasilinear elliptic equations with large exponents, J. Differ. Equ., 117 (1995), 28-55.  doi: 10.1006/jdeq.1995.1047.

[19]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[20]

G. Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

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