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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 and Applied Analysis, 2013, 12 (6) : 2445-2464. doi: 10.3934/cpaa.2013.12.2445
References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.

[3]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.

[4]

H. Brezis 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.

[5]

A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 463-470.

[6]

M. Comte, Solutions of elliptic equations with critical Sobolev exponent in dimension three, Nonlinear Anal., 17 (1991), 445-455. doi: 10.1016/0362-546X(91)90139-R.

[7]

A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034.

[8]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[9]

O. Druet, Elliptic equations with critical Sobolev exponents in dimension $3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 125-142.

[10]

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

[11]

F. Gazzola and B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differential Equations, 2 (1997), 555-572.

[12]

P. H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 215-223.

[13]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., 65, American Mathematical Society, Providence, RI, 1986.

[14]

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

[15]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, to appear in Rev. Mat. Iberoam., 29 (2013).

[16]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[17]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.

[18]

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

[19]

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

[20]

M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Springer Verlag, Berlin-Heidelberg, 1990.

[21]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41.

[22]

M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser, Boston, 1996.

show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.

[3]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.

[4]

H. Brezis 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.

[5]

A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 463-470.

[6]

M. Comte, Solutions of elliptic equations with critical Sobolev exponent in dimension three, Nonlinear Anal., 17 (1991), 445-455. doi: 10.1016/0362-546X(91)90139-R.

[7]

A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034.

[8]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[9]

O. Druet, Elliptic equations with critical Sobolev exponents in dimension $3$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 125-142.

[10]

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

[11]

F. Gazzola and B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differential Equations, 2 (1997), 555-572.

[12]

P. H. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 215-223.

[13]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., 65, American Mathematical Society, Providence, RI, 1986.

[14]

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

[15]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, to appear in Rev. Mat. Iberoam., 29 (2013).

[16]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[17]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.

[18]

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

[19]

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

[20]

M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Springer Verlag, Berlin-Heidelberg, 1990.

[21]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41.

[22]

M. Willem, "Minimax Theorems," Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser, Boston, 1996.

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