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

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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 and 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.
[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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