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

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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

Raffaella Servadei ^1, and Enrico Valdinoci ^2,

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

Abstract
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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.

Keywords: Critical nonlinearities, integrodifferential operators, variational techniques, fractional Laplacian., Mountain Pass Theorem, Linking Theorem, best critical Sobolev constant.

Mathematics Subject Classification: Primary: 49J35, 35A15, 35S15; Secondary: 47G20, 45G0.

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, available at http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=12-40.
[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, available at http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=12-58.
[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, available at http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=12-40.
[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, available at http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=12-58.
[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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