American Institute of Mathematical Sciences

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:
 [1] A. Aghajani, New a priori estimates for semistable solutions of semilinear elliptic equations, Potential Anal., 44 (2016), 729-744. [2] 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. [3] 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. [4] 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. [5] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Mat. Univ. Complut. Madrid, 10 (1997), 443-469. [6] X. Cabŕe, k-Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380. [7] 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. [8] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700. [9] 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 [10] 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. [11] 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. [12] 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. [13] 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. [14] 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 [15] L. Dupaigne, M. Ghergu and G. Warnault, The Gelfand Problem for the Biharmonic Operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752. [16] L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solution for the Liouville system, Geometric Partial Differential Equations, 208 (2013), 139-144. [17] 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. [18] A. Ferrero, H.-C. Grunau and P. Karageorgis, Supercritical biharmonic equations with power-type nonlinearity, Ann. Mat. Pura Appl., 188 (2009), 171-185. [19] N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449. [20] 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. [21] 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. [22] Z. Guo and J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054. [23] 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. [24] A. Moradifam, The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math. Soc., 138 (2010), 1287-1293. [25] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168. [26] 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. [27] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S'er. I Math., 330 (2000), 997-1002. [28] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302. [29] S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133. [30] K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260. [31] 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. [32] D. Ye and J. Wei, Liouville Theorems for finite Morse index solutions of Biharmonic problem, Math. Ann., 356 (2013), 1599-1612. [33] D. Ye and F. Zhou, Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math., 4 (2002), 547-558.

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References:
 [1] A. Aghajani, New a priori estimates for semistable solutions of semilinear elliptic equations, Potential Anal., 44 (2016), 729-744. [2] 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. [3] 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. [4] 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. [5] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Mat. Univ. Complut. Madrid, 10 (1997), 443-469. [6] X. Cabŕe, k-Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math., 63 (2010), 1362-1380. [7] 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. [8] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Adv. Nonlinear Stud., 11 (2011), 695-700. [9] 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 [10] 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. [11] 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. [12] 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. [13] 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. [14] 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 [15] L. Dupaigne, M. Ghergu and G. Warnault, The Gelfand Problem for the Biharmonic Operator, Arch. Ration. Mech. Anal., 208 (2013), 725-752. [16] L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solution for the Liouville system, Geometric Partial Differential Equations, 208 (2013), 139-144. [17] 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. [18] A. Ferrero, H.-C. Grunau and P. Karageorgis, Supercritical biharmonic equations with power-type nonlinearity, Ann. Mat. Pura Appl., 188 (2009), 171-185. [19] N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449. [20] 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. [21] 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. [22] Z. Guo and J. Wei, On a fourth order nonlinear elliptic equation with negative exponent, SIAM J. Math. Anal., 40 (2008/09), 2034-2054. [23] 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. [24] A. Moradifam, The singular extremal solutions of the bilaplacian with exponential nonlinearity, Proc. Amer. Math. Soc., 138 (2010), 1287-1293. [25] Y. Martel, Uniqueness of weak extremal solutions of nonlinear elliptic problems, Houston J. Math., 23 (1997), 161-168. [26] 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. [27] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris S'er. I Math., 330 (2000), 997-1002. [28] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302. [29] S. Villegas, Boundedness of extremal solutions in dimension 4, Adv. Math., 235 (2013), 126-133. [30] K. Wang, Partial regularity of stable solutions to the supercritical equations and its applications, Nonlinear Anal., 75 (2012), 5238-5260. [31] 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. [32] D. Ye and J. Wei, Liouville Theorems for finite Morse index solutions of Biharmonic problem, Math. Ann., 356 (2013), 1599-1612. [33] D. Ye and F. Zhou, Boundedness of the extremal solution for semilinear elliptic problems, Commun. Contemp. Math., 4 (2002), 547-558.
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