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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
##### References:
 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [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.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar [3] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar [4] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar [5] A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 2 (1985), 463.   Google Scholar [6] M. Comte, Solutions of elliptic equations with critical Sobolev exponent in dimension three,, Nonlinear Anal., 17 (1991), 445.  doi: 10.1016/0362-546X(91)90139-R.  Google Scholar [7] A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives,, J. Math. Anal. Appl., 295 (2004), 225.  doi: 10.1016/j.jmaa.2004.03.034.  Google Scholar [8] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar [9] O. Druet, Elliptic equations with critical Sobolev exponents in dimension $3$,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 19 (2002), 125.   Google Scholar [10] A. Fiscella, Saddle point solutions for non-local elliptic operators,, preprint., ().   Google Scholar [11] F. Gazzola and B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations,, Adv. Differential Equations, 2 (1997), 555.   Google Scholar [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.   Google Scholar [13] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Reg. Conf. Ser. Math.}, (1986).   Google Scholar [14] R. Servadei, The Yamabe equation in a non-local setting,, preprint, (): 12.   Google Scholar [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).   Google Scholar [16] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, J. Math. Anal. Appl., 389 (2012), 887.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar [17] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete Contin. Dyn. Syst., 33 (2013), 2105.  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] M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Ergebnisse der Mathematik und ihrer Grenzgebiete, (1990).   Google Scholar [21] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 36 (2011), 21.   Google Scholar [22] M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, (1996).   Google Scholar

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##### References:
 [1] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [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.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar [3] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar [4] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Comm. Pure Appl. Math., 36 (1983), 437.  doi: 10.1002/cpa.3160360405.  Google Scholar [5] A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 2 (1985), 463.   Google Scholar [6] M. Comte, Solutions of elliptic equations with critical Sobolev exponent in dimension three,, Nonlinear Anal., 17 (1991), 445.  doi: 10.1016/0362-546X(91)90139-R.  Google Scholar [7] A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives,, J. Math. Anal. Appl., 295 (2004), 225.  doi: 10.1016/j.jmaa.2004.03.034.  Google Scholar [8] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar [9] O. Druet, Elliptic equations with critical Sobolev exponents in dimension $3$,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 19 (2002), 125.   Google Scholar [10] A. Fiscella, Saddle point solutions for non-local elliptic operators,, preprint., ().   Google Scholar [11] F. Gazzola and B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations,, Adv. Differential Equations, 2 (1997), 555.   Google Scholar [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.   Google Scholar [13] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Reg. Conf. Ser. Math.}, (1986).   Google Scholar [14] R. Servadei, The Yamabe equation in a non-local setting,, preprint, (): 12.   Google Scholar [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).   Google Scholar [16] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, J. Math. Anal. Appl., 389 (2012), 887.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar [17] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete Contin. Dyn. Syst., 33 (2013), 2105.  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] M. Struwe, "Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems,", Ergebnisse der Mathematik und ihrer Grenzgebiete, (1990).   Google Scholar [21] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 36 (2011), 21.   Google Scholar [22] M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, (1996).   Google Scholar
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