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Oblique derivative problems for elliptic and parabolic equations
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 |
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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[9] |
O. Druet, Elliptic equations with critical Sobolev exponents in dimension $3$,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 19 (2002), 125.
|
[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.
|
[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.
|
[13] |
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Reg. Conf. Ser. Math.}, (1986).
|
[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. |
[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. |
[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).
|
[21] |
J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 36 (2011), 21.
|
[22] |
M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, (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.
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.
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.
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.
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. Poincar\'e Anal. Non Lin\'eaire, 2 (1985), 463.
|
[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. |
[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. |
[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. |
[9] |
O. Druet, Elliptic equations with critical Sobolev exponents in dimension $3$,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 19 (2002), 125.
|
[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.
|
[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.
|
[13] |
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations,, CBMS Reg. Conf. Ser. Math.}, (1986).
|
[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. |
[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. |
[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).
|
[21] |
J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 36 (2011), 21.
|
[22] |
M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, (1996).
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