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Existence results for the fractional Q-curvature problem on three dimensional CR sphere
Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran |
$Δ^{2}u = λ f(u)$ |
$Ω$ |
$R^{n}$ |
$u = Δ u = 0$ |
$\partial Ω$ |
$λ$ |
$ f:[0, a_{f}) \to \Bbb{R}_{+} $ |
$ \left( {0 < {a_f} \le \infty } \right)$ |
$ f(0) > 0 $ |
$ {a_f} $ |
$0<τ_{-}: = \liminf\limits_{t \to a_{f}} \frac{f(t)f''(t)}{f'(t)^{2}}≤q τ_{+}: = \limsup\limits_{t \to a_{f}} \frac{f(t)f''(t)}{f'(t)^{2}}<2.$ |
$\sqrt{λ_{m}}\int{{_{Ω}}}\sqrt{f'(u_{m})}\phi ^{2}dx≤\int{{_{Ω}}}|\nablaφ|^{2}dx, ~~\text{for all}~\phi ∈ H^{1}_{0}(Ω), $ |
$(2-τ_{-})^{2} α^{4}- 8(2-τ_{+})α^{2}+4(4-3τ_{+})α-4(1-τ_{+}) = 0.$ |
References:
[1] |
A. Aghajani,
New a priori estimates for semistable solutions of semilinear elliptic equations, Potential Anal., 44 (2016), 729-744.
|
[2] |
A. Aghajani,
Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains, Discrete Contin. Dyn. Syst., 37 (2017), 3521-3530.
|
[3] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.
|
[4] |
E. Berchio and F. Gazoola, Some remarks on bihormonic elliptic problems with positive, increasing and convex nonlinearities,
Electronic J. differential Equations, 34 (2005), 20 pp. |
[5] |
H. Brezis and L. Vazquez,
Blow-up solutions of some nonlinear elliptic problems, Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[6] |
X. Cabŕe,
k-Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.
|
[7] |
D. Cassani, J. do O and N. Ghoussoub,
On a fourth order elliptic problem with a singular nonlinearity, Adv. Nonlinear Stud., 9 (2009), 177-197.
|
[8] |
C. Cowan,
Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700.
|
[9] |
C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity,
Arch. Ration. Mech. Anal., in press, (2009), 19 pp |
[10] |
C. Cowan, P. Esposito and N. Ghoussoub,
Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, DCDS-A, 28 (2010), 1033-1050.
|
[11] |
C. Cowan and N. Ghoussoub,
Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Cal. Var., 49 (2014), 291-305.
|
[12] |
X. Cabŕe, M. Sanchón and J. Spruck,
A priori estimates for semistable solutions of semilinear elliptic equations, Discrete Contin. Dyn. Syst., 39 (2007), 565-592.
|
[13] |
J. Dávila, L. Dupaigne, I. Guerra and M. Montenegro,
Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592.
|
[14] |
J. Dávila, I. Flores and I. Guerra, Multiplicity of solutions for a fourth order equation with power-type nonlinearity,
Math. Ann., 348 (2010), 143--193 |
[15] |
L. Dupaigne, M. Ghergu and G. Warnault,
The Gelfand Problem for the Biharmonic Operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752.
|
[16] |
L. Dupaigne, A. Farina and B. Sirakov,
Regularity of the extremal solution for the Liouville system, Geometric Partial Differential Equations, 208 (2013), 139-144.
|
[17] |
P. Esposito, N. Ghoussoub and Y. Guo,
Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768.
|
[18] |
A. Ferrero, H.-C. Grunau and P. Karageorgis,
Supercritical biharmonic equations with power-type nonlinearity, Ann. Mat. Pura Appl., 188 (2009), 171-185.
|
[19] |
N. Ghoussoub and Y. Guo,
On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449.
|
[20] |
Z. Guo and J. Wei,
Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580.
|
[21] |
F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, (1991), Springer, Berlin, 2010. |
[22] |
Z. Guo and J. Wei,
On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054.
|
[23] |
H. Hajlaoui, A. Harrabi and D. Ye,
On stable solutions of the biharmonic problem with polynomial growth, Pacific Journal of Mathematics, 270 (2014), 79-93.
|
[24] |
A. Moradifam,
The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math. Soc., 138 (2010), 1287-1293.
|
[25] |
Y. Martel,
Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168.
|
[26] |
F. Mignot and J-P. Puel,
Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836.
|
[27] |
G. Nedev,
Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S'er. I Math., 330 (2000), 997-1002.
|
[28] |
J. Serrin,
Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.
|
[29] |
S. Villegas,
Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.
|
[30] |
K. Wang,
Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260.
|
[31] |
J. Wei, X. Xu and W. Yang,
On the classification of stable solutions to biharmonic problems in large dimensions, Pacific J. Math., 263 (2013), 495-512.
|
[32] |
D. Ye and J. Wei,
Liouville Theorems for finite Morse index solutions of Biharmonic problem, Math. Ann., 356 (2013), 1599-1612.
|
[33] |
D. Ye and F. Zhou,
Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math., 4 (2002), 547-558.
|
show all references
References:
[1] |
A. Aghajani,
New a priori estimates for semistable solutions of semilinear elliptic equations, Potential Anal., 44 (2016), 729-744.
|
[2] |
A. Aghajani,
Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains, Discrete Contin. Dyn. Syst., 37 (2017), 3521-3530.
|
[3] |
S. Agmon, A. Douglis and L. Nirenberg,
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.
|
[4] |
E. Berchio and F. Gazoola, Some remarks on bihormonic elliptic problems with positive, increasing and convex nonlinearities,
Electronic J. differential Equations, 34 (2005), 20 pp. |
[5] |
H. Brezis and L. Vazquez,
Blow-up solutions of some nonlinear elliptic problems, Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[6] |
X. Cabŕe,
k-Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.
|
[7] |
D. Cassani, J. do O and N. Ghoussoub,
On a fourth order elliptic problem with a singular nonlinearity, Adv. Nonlinear Stud., 9 (2009), 177-197.
|
[8] |
C. Cowan,
Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700.
|
[9] |
C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity,
Arch. Ration. Mech. Anal., in press, (2009), 19 pp |
[10] |
C. Cowan, P. Esposito and N. Ghoussoub,
Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, DCDS-A, 28 (2010), 1033-1050.
|
[11] |
C. Cowan and N. Ghoussoub,
Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Cal. Var., 49 (2014), 291-305.
|
[12] |
X. Cabŕe, M. Sanchón and J. Spruck,
A priori estimates for semistable solutions of semilinear elliptic equations, Discrete Contin. Dyn. Syst., 39 (2007), 565-592.
|
[13] |
J. Dávila, L. Dupaigne, I. Guerra and M. Montenegro,
Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592.
|
[14] |
J. Dávila, I. Flores and I. Guerra, Multiplicity of solutions for a fourth order equation with power-type nonlinearity,
Math. Ann., 348 (2010), 143--193 |
[15] |
L. Dupaigne, M. Ghergu and G. Warnault,
The Gelfand Problem for the Biharmonic Operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752.
|
[16] |
L. Dupaigne, A. Farina and B. Sirakov,
Regularity of the extremal solution for the Liouville system, Geometric Partial Differential Equations, 208 (2013), 139-144.
|
[17] |
P. Esposito, N. Ghoussoub and Y. Guo,
Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768.
|
[18] |
A. Ferrero, H.-C. Grunau and P. Karageorgis,
Supercritical biharmonic equations with power-type nonlinearity, Ann. Mat. Pura Appl., 188 (2009), 171-185.
|
[19] |
N. Ghoussoub and Y. Guo,
On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449.
|
[20] |
Z. Guo and J. Wei,
Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580.
|
[21] |
F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, (1991), Springer, Berlin, 2010. |
[22] |
Z. Guo and J. Wei,
On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054.
|
[23] |
H. Hajlaoui, A. Harrabi and D. Ye,
On stable solutions of the biharmonic problem with polynomial growth, Pacific Journal of Mathematics, 270 (2014), 79-93.
|
[24] |
A. Moradifam,
The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math. Soc., 138 (2010), 1287-1293.
|
[25] |
Y. Martel,
Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168.
|
[26] |
F. Mignot and J-P. Puel,
Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836.
|
[27] |
G. Nedev,
Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S'er. I Math., 330 (2000), 997-1002.
|
[28] |
J. Serrin,
Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.
|
[29] |
S. Villegas,
Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.
|
[30] |
K. Wang,
Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260.
|
[31] |
J. Wei, X. Xu and W. Yang,
On the classification of stable solutions to biharmonic problems in large dimensions, Pacific J. Math., 263 (2013), 495-512.
|
[32] |
D. Ye and J. Wei,
Liouville Theorems for finite Morse index solutions of Biharmonic problem, Math. Ann., 356 (2013), 1599-1612.
|
[33] |
D. Ye and F. Zhou,
Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math., 4 (2002), 547-558.
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