We study the regularity of the extremal solution of the semilinear biharmonic equation $\Delta^2 u = \frac{\lambda}{(1-u)^}$, which models a simple micro-electromechanical system (MEMS) device on a ball $B \subset R^N$, under Dirichlet boundary conditions $u = \partial_\nu u =0$ on ∂B. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” $λ∗ > 0$ such that a stable classical solution $u_λ$ with $0 < u_λ < 1$ exists for λ ∈ (0,λ∗), while there is none of any kind when λ > λ∗. Our main result asserts that the extremal solution u_λ∗ is regular (sup_B u_λ∗ < 1) provided N\leq 8 while u_λ∗ is singular (sup_B u_λ∗ = 1) for N\geq 9, in which case $1−C_0 |x|^{4/3}\leq u_λ∗(x)\leq 1 − |x|^{4/3}$ on the unit ball, where $C 0= (λ∗/\bar λ)^{1/3} and $\bar λ=8/9 (N−2/3)(N−8/3)$.

Cowan, C., Esposito, P., Ghoussoub, N., Moradifam, A. (2010). The critical dimension for a fourth order elliptic problem with singular nonlinearity. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 198(3), 763-787 [10.1007/s00205-010-0367-x].

The critical dimension for a fourth order elliptic problem with singular nonlinearity

ESPOSITO, PIERPAOLO;
2010-01-01

Abstract

We study the regularity of the extremal solution of the semilinear biharmonic equation $\Delta^2 u = \frac{\lambda}{(1-u)^}$, which models a simple micro-electromechanical system (MEMS) device on a ball $B \subset R^N$, under Dirichlet boundary conditions $u = \partial_\nu u =0$ on ∂B. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” $λ∗ > 0$ such that a stable classical solution $u_λ$ with $0 < u_λ < 1$ exists for λ ∈ (0,λ∗), while there is none of any kind when λ > λ∗. Our main result asserts that the extremal solution u_λ∗ is regular (sup_B u_λ∗ < 1) provided N\leq 8 while u_λ∗ is singular (sup_B u_λ∗ = 1) for N\geq 9, in which case $1−C_0 |x|^{4/3}\leq u_λ∗(x)\leq 1 − |x|^{4/3}$ on the unit ball, where $C 0= (λ∗/\bar λ)^{1/3} and $\bar λ=8/9 (N−2/3)(N−8/3)$.
2010
Cowan, C., Esposito, P., Ghoussoub, N., Moradifam, A. (2010). The critical dimension for a fourth order elliptic problem with singular nonlinearity. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 198(3), 763-787 [10.1007/s00205-010-0367-x].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/142527
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