# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [1] Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021036 [2] Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021041 [3] Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021026 [4] Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031 [5] Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002 [6] Emily McMillon, Allison Beemer, Christine A. Kelley. Extremal absorbing sets in low-density parity-check codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021003 [7] Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017 [8] Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 [9] Francesca Bucci. Improved boundary regularity for a Stokes-Lamé system. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021018 [10] Leon Mons. Partial regularity for parabolic systems with VMO-coefficients. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021041 [11] Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099 [12] Mengjie Zhang. Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021038 [13] Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024 [14] Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341 [15] Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 [16] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [17] Melis Alpaslan Takan, Refail Kasimbeyli. Multiobjective mathematical models and solution approaches for heterogeneous fixed fleet vehicle routing problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2073-2095. doi: 10.3934/jimo.2020059 [18] Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $BV$ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405 [19] Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023 [20] Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021

2019 Impact Factor: 1.105