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November  2013, 12(6): 2445-2464. doi: 10.3934/cpaa.2013.12.2445

A Brezis-Nirenberg result for non-local critical equations in low dimension

1. 

Dipartimento di Matematica, Università degli Studi della Calabria, Ponte Pietro Bucci 31B, I–87036 Arcavacata di Rende

2. 

Università di Roma Tor Vergata, Dipartimento di Matematica, Via della Ricerca Scientifica, I-00133 Rome

Received  April 2012 Revised  January 2013 Published  May 2013

The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities \begin{eqnarray} (-\Delta)^s u-\lambda u=|u|^{2^*-2}u, in \Omega \\ u=0, in R^n\setminus \Omega, \end{eqnarray} where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $R^n$, $n>2s$, with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when $\Omega$ is an open bounded subset of $R^n$ with $n\geq 4s$ and, in this framework, we prove some existence results. Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when $2s < n < 4s$. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension. In particular when $s=1$ (and consequently $n=3$) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4]. In this sense the present work may be considered as the extension of some classical results for the Laplacian to the case of non-local fractional operators.
Citation: Raffaella Servadei, Enrico Valdinoci. A Brezis-Nirenberg result for non-local critical equations in low dimension. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2445-2464. doi: 10.3934/cpaa.2013.12.2445
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show all references

References:
[1]

J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.  Google Scholar

[3]

Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[4]

Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405.  Google Scholar

[5]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 463-470.  Google Scholar

[6]

Nonlinear Anal., 17 (1991), 445-455. doi: 10.1016/0362-546X(91)90139-R.  Google Scholar

[7]

J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034.  Google Scholar

[8]

Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[9]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 125-142.  Google Scholar

[10]

A. Fiscella, Saddle point solutions for non-local elliptic operators,, preprint., ().   Google Scholar

[11]

Adv. Differential Equations, 2 (1997), 555-572.  Google Scholar

[12]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 215-223.  Google Scholar

[13]

CBMS Reg. Conf. Ser. Math., 65, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[14]

R. Servadei, The Yamabe equation in a non-local setting,, preprint, (): 12.   Google Scholar

[15]

to appear in Rev. Mat. Iberoam., 29 (2013). Google Scholar

[16]

J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[17]

Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[18]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, to appear in Trans. Amer. Math. Soc., ().   Google Scholar

[19]

R. Servadei and E. Valdinoci, Fractional Laplacian equations with critical Sobolev exponent,, preprint, (): 12.   Google Scholar

[20]

Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Springer Verlag, Berlin-Heidelberg, 1990.  Google Scholar

[21]

Calc. Var. Partial Differential Equations, 36 (2011), 21-41.  Google Scholar

[22]

Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser, Boston, 1996.  Google Scholar

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