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August  2017, 37(7): 3521-3530. doi: 10.3934/dcds.2017150

Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains

Asadollah Aghajani 1,2,

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

Received  May 2016 Revised  March 2017 Published  April 2017

Fund Project: The author is supported by IPM grant 95340123.

We consider the semilinear elliptic equation
$ -\Delta u =\lambda f(u) $
 in a smooth bounded domain
$ \Omega $
 of
$ \Bbb{R}^{n} $
 with Dirichlet boundary condition, where
$ f $
 is a
$ C^{1} $
 positive and nondeccreasing function in
$ [0, \infty) $
 such that
$ \frac{f(t)}{t} \rightarrow \infty $
 as
$ t \rightarrow \infty $
. When
$ \Omega $
 is an arbitrary domain and
$ f $
 is not necessarily convex, the boundedness of the extremal solution
$ u^{*} $
 is known only for
$ n = 2 $
, established by X. Cabré[5]. In this paper, we prove this for higher dimensions depending on the nonlinearity
$ f $
. In particular, we prove that if
$ \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, $
where
$ F(t)=\int_{0}^{t}f(s)ds $
, then
$ u^{*} \in L^{\infty}(\Omega) $
, for
$ n \leq 6 $
. Also, if
$\beta_{-}=\beta_{+}>\frac{1}{2} $
 or
$ \frac{1}{2} < \beta_{-}\leq \beta_{+} < \frac{7}{10} $
, then
$ u^{*} \in L^{\infty}(\Omega) $
, for
$ n \leq 9 $
. Moreover, under the sole condition that
$ \beta_{-} > \frac{1}{2} $
 we have
$ u^{*} \in H^{1}_{0}(\Omega) $
 for
$ n \geq 1 $
. The same is true if for some
$ \epsilon > 0 $
 we have
$$$ \frac{tf'(t)}{f(t)} \geq 1+\frac{1}{(\ln t)^{2-\epsilon}} ~~ \text{for large} ~ t, $$$
which improves a similar result by Brezis and Vázquez [4].
Keywords: Extremal solutions, boundedness, semilinear elliptic equations..
Mathematics Subject Classification: Primary: 35K57, 35B65; Secondary: 35J60.
Citation: Asadollah Aghajani. Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3521-3530. doi: 10.3934/dcds.2017150
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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.  doi: 10.1007/BF02391014.  Google Scholar

[20]

N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308.   Google Scholar

[21]

S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.  doi: 10.1016/j.aim.2012.11.015.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.  doi: 10.1007/BF02391014.  Google Scholar

[20]

N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308.   Google Scholar

[21]

S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133.  doi: 10.1016/j.aim.2012.11.015.  Google Scholar

[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.  Google Scholar

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