    May  2018, 17(3): 887-898. doi: 10.3934/cpaa.2018044

## Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran

* Corresponding author: A. Aghajani

Received  August 2017 Revised  October 2017 Published  January 2018

We consider the fourth order problem
 $Δ^{2}u = λ f(u)$
on a general bounded domain
 $Ω$
in
 $R^{n}$
with the Navier boundary condition
 $u = Δ u = 0$
on
 $\partial Ω$
. Here,
 $λ$
is a positive parameter and
 $f:[0, a_{f}) \to \Bbb{R}_{+}$
 $\left( {0 < {a_f} \le \infty } \right)$
is a smooth, increasing, convex nonlinearity such that
 $f(0) > 0$
and which blows up at
 ${a_f}$
. Let
 $0<τ_{-}: = \liminf\limits_{t \to a_{f}} \frac{f(t)f''(t)}{f'(t)^{2}}≤q τ_{+}: = \limsup\limits_{t \to a_{f}} \frac{f(t)f''(t)}{f'(t)^{2}}<2.$
We show that if $u_{m}$ is a sequence of semistable solutions correspond to $λ_{m}$ satisfy the stability inequality
 $\sqrt{λ_{m}}\int{{_{Ω}}}\sqrt{f'(u_{m})}\phi ^{2}dx≤\int{{_{Ω}}}|\nablaφ|^{2}dx, ~~\text{for all}~\phi ∈ H^{1}_{0}(Ω),$
then $\sup_{m} ||u_{m}||_{L^{∞}(Ω)}<a_{f}$ for $n< \frac{4α_{*}(2-τ_{+})+2τ_{+}}{τ_{+}}\max \{1, τ_{+}\},$ where $α^{*}$ is the largest root of the equation
 $(2-τ_{-})^{2} α^{4}- 8(2-τ_{+})α^{2}+4(4-3τ_{+})α-4(1-τ_{+}) = 0.$
In particular, if $τ_{-} = τ_{+}: = τ$, then $\sup_{m} ||u_{m}||_{L^{∞}(Ω)}<a_{f}$ for $n≤12$ when $τ≤ 1$, and for $n≤7$ when $τ≤ 1.57863$. These estimates lead to the regularity of the corresponding extremal solution $u^{*}(x) = \lim_{λ\uparrowλ^{*}}u_{λ}(x),$ where $λ^*$ is the extremal parameter of the eigenvalue problem.
Citation: A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044
##### References:
  A. Aghajani, New a priori estimates for semistable solutions of semilinear elliptic equations, Potential Anal., 44 (2016), 729-744. Google Scholar  A. Aghajani, Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains, Discrete Contin. Dyn. Syst., 37 (2017), 3521-3530. Google Scholar  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. Google Scholar  E. Berchio and F. Gazoola, Some remarks on bihormonic elliptic problems with positive, increasing and convex nonlinearities, Electronic J. differential Equations, 34 (2005), 20 pp. Google Scholar  H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Mat. Univ. Complut. Madrid, 10 (1997), 443-469. Google Scholar  X. Cabŕe, k-Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380. Google Scholar  D. Cassani, J. do O and N. Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity, Adv. Nonlinear Stud., 9 (2009), 177-197. Google Scholar  C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700. Google Scholar  C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., in press, (2009), 19 pp Google Scholar  C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, DCDS-A, 28 (2010), 1033-1050. Google Scholar  C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Cal. Var., 49 (2014), 291-305. Google Scholar  X. Cabŕe, M. Sanchón and J. Spruck, A priori estimates for semistable solutions of semilinear elliptic equations, Discrete Contin. Dyn. Syst., 39 (2007), 565-592. Google Scholar  J. Dávila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592. Google Scholar  J. Dávila, I. Flores and I. Guerra, Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348 (2010), 143--193 Google Scholar  L. Dupaigne, M. Ghergu and G. Warnault, The Gelfand Problem for the Biharmonic Operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752. Google Scholar  L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solution for the Liouville system, Geometric Partial Differential Equations, 208 (2013), 139-144. Google Scholar  P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768. Google Scholar  A. Ferrero, H.-C. Grunau and P. Karageorgis, Supercritical biharmonic equations with power-type nonlinearity, Ann. Mat. Pura Appl., 188 (2009), 171-185. Google Scholar  N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449. Google Scholar  Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580. Google Scholar  F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, (1991), Springer, Berlin, 2010. Google Scholar  Z. Guo and J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054. Google Scholar  H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of the biharmonic problem with polynomial growth, Pacific Journal of Mathematics, 270 (2014), 79-93. Google Scholar  A. Moradifam, The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math. Soc., 138 (2010), 1287-1293. Google Scholar  Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168. Google Scholar  F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836. Google Scholar  G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S'er. I Math., 330 (2000), 997-1002. Google Scholar  J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302. Google Scholar  S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133. Google Scholar  K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260. Google Scholar  J. Wei, X. Xu and W. Yang, On the classification of stable solutions to biharmonic problems in large dimensions, Pacific J. Math., 263 (2013), 495-512. Google Scholar  D. Ye and J. Wei, Liouville Theorems for finite Morse index solutions of Biharmonic problem, Math. Ann., 356 (2013), 1599-1612. Google Scholar  D. Ye and F. Zhou, Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math., 4 (2002), 547-558. Google Scholar

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##### References:
  A. Aghajani, New a priori estimates for semistable solutions of semilinear elliptic equations, Potential Anal., 44 (2016), 729-744. Google Scholar  A. Aghajani, Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains, Discrete Contin. Dyn. Syst., 37 (2017), 3521-3530. Google Scholar  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. Google Scholar  E. Berchio and F. Gazoola, Some remarks on bihormonic elliptic problems with positive, increasing and convex nonlinearities, Electronic J. differential Equations, 34 (2005), 20 pp. Google Scholar  H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Mat. Univ. Complut. Madrid, 10 (1997), 443-469. Google Scholar  X. Cabŕe, k-Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380. Google Scholar  D. Cassani, J. do O and N. Ghoussoub, On a fourth order elliptic problem with a singular nonlinearity, Adv. Nonlinear Stud., 9 (2009), 177-197. Google Scholar  C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700. Google Scholar  C. Cowan, P. Esposito, N. Ghoussoub and A. Moradifam, The critical dimension for a fourth order elliptic problem with singular nonlinearity, Arch. Ration. Mech. Anal., in press, (2009), 19 pp Google Scholar  C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, DCDS-A, 28 (2010), 1033-1050. Google Scholar  C. Cowan and N. Ghoussoub, Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains, Cal. Var., 49 (2014), 291-305. Google Scholar  X. Cabŕe, M. Sanchón and J. Spruck, A priori estimates for semistable solutions of semilinear elliptic equations, Discrete Contin. Dyn. Syst., 39 (2007), 565-592. Google Scholar  J. Dávila, L. Dupaigne, I. Guerra and M. Montenegro, Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal., 39 (2007), 565-592. Google Scholar  J. Dávila, I. Flores and I. Guerra, Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348 (2010), 143--193 Google Scholar  L. Dupaigne, M. Ghergu and G. Warnault, The Gelfand Problem for the Biharmonic Operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752. Google Scholar  L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solution for the Liouville system, Geometric Partial Differential Equations, 208 (2013), 139-144. Google Scholar  P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768. Google Scholar  A. Ferrero, H.-C. Grunau and P. Karageorgis, Supercritical biharmonic equations with power-type nonlinearity, Ann. Mat. Pura Appl., 188 (2009), 171-185. Google Scholar  N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449. Google Scholar  Z. Guo and J. Wei, Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents, Discrete Contin. Dyn. Syst., 34 (2014), 2561-2580. Google Scholar  F. Gazzola, H. -C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Mathematics, (1991), Springer, Berlin, 2010. Google Scholar  Z. Guo and J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054. Google Scholar  H. Hajlaoui, A. Harrabi and D. Ye, On stable solutions of the biharmonic problem with polynomial growth, Pacific Journal of Mathematics, 270 (2014), 79-93. Google Scholar  A. Moradifam, The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math. Soc., 138 (2010), 1287-1293. Google Scholar  Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168. Google Scholar  F. Mignot and J-P. Puel, Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836. Google Scholar  G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S'er. I Math., 330 (2000), 997-1002. Google Scholar  J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302. Google Scholar  S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133. Google Scholar  K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260. Google Scholar  J. Wei, X. Xu and W. Yang, On the classification of stable solutions to biharmonic problems in large dimensions, Pacific J. Math., 263 (2013), 495-512. Google Scholar  D. Ye and J. Wei, Liouville Theorems for finite Morse index solutions of Biharmonic problem, Math. Ann., 356 (2013), 1599-1612. Google Scholar  D. Ye and F. Zhou, Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math., 4 (2002), 547-558. Google Scholar
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