American Institute of Mathematical Sciences

Advanced
July  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 - A, 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

[1]

Xavier Cabré. A new proof of the boundedness results for stable solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7249-7264. doi: 10.3934/dcds.2019302
[2]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601
[3]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
[4]

Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886
[5]

David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335
[6]

Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961
[7]

Mostafa Fazly. Regularity of extremal solutions of nonlocal elliptic systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 107-131. doi: 10.3934/dcds.2020005
[8]

Zhuoran Du. Some properties of positive radial solutions for some semilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 943-953. doi: 10.3934/cpaa.2010.9.943
[9]

Tomás Sanz-Perela. Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2547-2575. doi: 10.3934/cpaa.2018121
[10]

Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50
[11]

Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044
[12]

Soohyun Bae. Classification of positive solutions of semilinear elliptic equations with Hardy term. Conference Publications, 2013, 2013 (special) : 31-39. doi: 10.3934/proc.2013.2013.31
[13]

Alan V. Lair, Ahmed Mohammed. Entire large solutions of semilinear elliptic equations of mixed type. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1607-1618. doi: 10.3934/cpaa.2009.8.1607
[14]

Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1381-1392. doi: 10.3934/cpaa.2013.12.1381
[15]

Sara Barile, Addolorata Salvatore. Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains. Conference Publications, 2013, 2013 (special) : 41-49. doi: 10.3934/proc.2013.2013.41
[16]

Zhijun Zhang, Ling Mi. Blow-up rates of large solutions for semilinear elliptic equations. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1733-1745. doi: 10.3934/cpaa.2011.10.1733
[17]

Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198
[18]

Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166
[19]

Jinlong Bai, Desheng Li, Chunqiu Li. A note on multiplicity of solutions near resonance of semilinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3351-3365. doi: 10.3934/cpaa.2019151
[20]

Francesca Alessio, Piero Montecchiari, Andrea Sfecci. Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations. Networks & Heterogeneous Media, 2019, 14 (3) : 567-587. doi: 10.3934/nhm.2019022

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (14)
  • HTML views (4)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences