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Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains
1. | School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran |
2. | School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, P.O.Box: 19395-5746, Iran |
$ -\Delta u =\lambda f(u) $ |
$ \Omega $ |
$ \Bbb{R}^{n} $ |
$ f $ |
$ C^{1} $ |
$ [0, \infty) $ |
$ \frac{f(t)}{t} \rightarrow \infty $ |
$ t \rightarrow \infty $ |
$ \Omega $ |
$ f $ |
$ u^{*} $ |
$ n = 2 $ |
$ f $ |
$ \frac{1}{2} < \beta_{-}:=\liminf\limits_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}\leq \beta_{+}:=\limsup\limits_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}} < \infty, $ |
$ F(t)=\int_{0}^{t}f(s)ds $ |
$ u^{*} \in L^{\infty}(\Omega) $ |
$ n \leq 6 $ |
$\beta_{-}=\beta_{+}>\frac{1}{2} $ |
$ \frac{1}{2} < \beta_{-}\leq \beta_{+} < \frac{7}{10} $ |
$ u^{*} \in L^{\infty}(\Omega) $ |
$ n \leq 9 $ |
$ \beta_{-} > \frac{1}{2} $ |
$ u^{*} \in H^{1}_{0}(\Omega) $ |
$ n \geq 1 $ |
$ \epsilon > 0 $ |
$$$ \frac{tf'(t)}{f(t)} \geq 1+\frac{1}{(\ln t)^{2-\epsilon}} ~~ \text{for large} ~ t, $$$ |
References:
[1] |
A. Aghajani,
New a priori estimates for semistable solutions of semilinear elliptic equations, Potential Anal., 44 (2016), 729-744.
doi: 10.1007/s11118-015-9528-8. |
[2] |
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.
doi: 10.1002/cpa.3160120405. |
[3] |
H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa,
Blow-up for $u_{t-\Delta u = g(u)}$ revisited, Adv. Differental Equation, 1 (1996), 73-90.
|
[4] |
H. Brezis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[5] |
X. Cabré,
Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.
doi: 10.1002/cpa.20327. |
[6] |
X. Cabré and A. Capella,
Regularity of radial minimizers and extremal solutions of semi-linear elliptic equations, J. Funct. Anal., 238 (2006), 709-733.
doi: 10.1016/j.jfa.2005.12.018. |
[7] |
X. Cabré, A. Capella and M. Sanchéon,
Regularity of radial minimizers of reaction equations involving the $ p $-Laplacian, Calc. Var. Partial Differential Equations, 34 (2009), 475-494.
doi: 10.1007/s00526-008-0192-3. |
[8] |
X. Cabré and X. Ros-Oton,
Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations, 38 (2013), 135-154.
doi: 10.1080/03605302.2012.697505. |
[9] |
X. Cabré and M. Sanchéon,
Geometric-type Hardy-Sobolev inequalities and applications to regularity of minimizers, J. Funct. Anal., 264 (2013), 303-325.
doi: 10.1016/j.jfa.2012.10.012. |
[10] |
X. Cabré, M. Sanchéon and J. Spruck,
A priori estimates for semistable solutions of semilinear elliptic equations, Discrete Contin. Dyn. Syst.. Series A, 36 (2016), 601-609.
doi: 10.3934/dcds.2016.36.601. |
[11] |
M. G. Crandall and P. H. Rabinowitz,
Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), 207-218.
doi: 10.1007/BF00280741. |
[12] |
J. Dávila, L. Dupaigne and M. Montenegro,
The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal., 7 (2008), 795-817.
doi: 10.3934/cpaa.2008.7.795. |
[13] |
L. Dupaigne,
Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, (2011).
doi: 10.1201/b10802. |
[14] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal., 49 (1973), 241-269.
doi: 10.1007/BF00250508. |
[15] |
F Mignot and J.-P. Puel,
Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836.
doi: 10.1080/03605308008820155. |
[16] |
G. Nedev,
Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 330 (2000), 997-1002.
doi: 10.1016/S0764-4442(00)00289-5. |
[17] |
G. Nedev,
Extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S'er. Ⅰ Math., 330 (2000), 997-1002.
doi: 10.1016/S0764-4442(00)00289-5. |
[18] |
M. Sanchéon,
Boundedness of the extremal solution of some $p$-Laplacian problems, Nonlinear Anal., 67 (2007), 281-294.
doi: 10.1016/j.na.2006.05.010. |
[19] |
J. Serrin,
Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.
doi: 10.1007/BF02391014. |
[20] |
N. S. Trudinger,
Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308.
|
[21] |
S. Villegas,
Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.
doi: 10.1016/j.aim.2012.11.015. |
[22] |
D. Ye and F. Zhou,
Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math., 4 (2002), 547-558.
doi: 10.1142/S0219199702000701. |
show all references
References:
[1] |
A. Aghajani,
New a priori estimates for semistable solutions of semilinear elliptic equations, Potential Anal., 44 (2016), 729-744.
doi: 10.1007/s11118-015-9528-8. |
[2] |
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.
doi: 10.1002/cpa.3160120405. |
[3] |
H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa,
Blow-up for $u_{t-\Delta u = g(u)}$ revisited, Adv. Differental Equation, 1 (1996), 73-90.
|
[4] |
H. Brezis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[5] |
X. Cabré,
Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380.
doi: 10.1002/cpa.20327. |
[6] |
X. Cabré and A. Capella,
Regularity of radial minimizers and extremal solutions of semi-linear elliptic equations, J. Funct. Anal., 238 (2006), 709-733.
doi: 10.1016/j.jfa.2005.12.018. |
[7] |
X. Cabré, A. Capella and M. Sanchéon,
Regularity of radial minimizers of reaction equations involving the $ p $-Laplacian, Calc. Var. Partial Differential Equations, 34 (2009), 475-494.
doi: 10.1007/s00526-008-0192-3. |
[8] |
X. Cabré and X. Ros-Oton,
Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations, 38 (2013), 135-154.
doi: 10.1080/03605302.2012.697505. |
[9] |
X. Cabré and M. Sanchéon,
Geometric-type Hardy-Sobolev inequalities and applications to regularity of minimizers, J. Funct. Anal., 264 (2013), 303-325.
doi: 10.1016/j.jfa.2012.10.012. |
[10] |
X. Cabré, M. Sanchéon and J. Spruck,
A priori estimates for semistable solutions of semilinear elliptic equations, Discrete Contin. Dyn. Syst.. Series A, 36 (2016), 601-609.
doi: 10.3934/dcds.2016.36.601. |
[11] |
M. G. Crandall and P. H. Rabinowitz,
Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), 207-218.
doi: 10.1007/BF00280741. |
[12] |
J. Dávila, L. Dupaigne and M. Montenegro,
The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal., 7 (2008), 795-817.
doi: 10.3934/cpaa.2008.7.795. |
[13] |
L. Dupaigne,
Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, (2011).
doi: 10.1201/b10802. |
[14] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal., 49 (1973), 241-269.
doi: 10.1007/BF00250508. |
[15] |
F Mignot and J.-P. Puel,
Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836.
doi: 10.1080/03605308008820155. |
[16] |
G. Nedev,
Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 330 (2000), 997-1002.
doi: 10.1016/S0764-4442(00)00289-5. |
[17] |
G. Nedev,
Extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S'er. Ⅰ Math., 330 (2000), 997-1002.
doi: 10.1016/S0764-4442(00)00289-5. |
[18] |
M. Sanchéon,
Boundedness of the extremal solution of some $p$-Laplacian problems, Nonlinear Anal., 67 (2007), 281-294.
doi: 10.1016/j.na.2006.05.010. |
[19] |
J. Serrin,
Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.
doi: 10.1007/BF02391014. |
[20] |
N. S. Trudinger,
Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308.
|
[21] |
S. Villegas,
Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.
doi: 10.1016/j.aim.2012.11.015. |
[22] |
D. Ye and F. Zhou,
Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math., 4 (2002), 547-558.
doi: 10.1142/S0219199702000701. |
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