# American Institute of Mathematical Sciences

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

 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]. 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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