On Lane–Emden Systems with Singular Nonlinearities and Applications to MEMS

1.1 Motivation and Related Results

System (Sλ,μ) can be seen as the Lane–Emden type system with nonlinearities with negative exponents (see e.g. [18, 28, 34, 35]). A lot of work has been devoted to existence and non-existence of solutions of elliptic systems with continuous power like nonlinearities, among which we recall [11, 12, 13, 14, 15, 26, 30] and the survey [10]. For a recent account of the Lane–Emden type singular nonlinearities we refer the reader to [22]. Here we address the problem of studying existence, non-existence and regularity results by means of the nonlinear eigenvalue problem (Sλ,μ), in which for the sake of clarity we consider a Coulomb nonlinear source though most results extend to more general situations. Related results for systems with continuous nonlinearities have been obtained in [25, 31].

Another important motivation to consider (Sλ,μ) comes from recent works on the study of the equations that model micro-electromechanical systems (MEMS):

MEMS are often used to combine electronics with micro-size mechanical devices in the design of various types of microscopic machinery. MEMS devices have therefore become key components of many commercial systems, including accelerometers for airbag deployment in vehicles, ink jet printer heads, optical switches and chemical sensors.

Nonlinear interaction described in terms of coupling of semilinear elliptic equations has revealed through the last decades a fundamental tool in studying nonlinear phenomena (see e.g. [3, 9, 13, 14, 16] and references therein). In all the above contexts the nonlinearity is fairly represented by a continuous function. More recently, a rigorous mathematical approach in modeling and designing micro electro mechanical systems has demanded the need to consider also nonlinearities which develop singularities. In a nutshell, one may think of MEMS’ actuation as governed by the dynamic of a micro plate which deflects towards a fixed plate, under the effect of a Coulomb force, once that a drop voltage is applied.

In the stationary case, the naive model which describes this device cast into the second order elliptic PDE (Pλ), where Ω is a bounded smooth domain in ℝN and the positive function g is bounded and related to dielectric properties of the material, see the survey [20] and also [23, 27, 32] for more technical aspects. The key feature of the equation in (Pλ) is retained by the discontinuity of the nonlinearity which blows up as v→1- and this corresponds in applications to a snap through of the device.

The general goal on the study of (Pλ) is to analyze the structure of the branch of solutions as well as their qualitative properties. The role of the positive parameter λ is that of tuning the drop voltage, whence from the PDE point of view, it yields the threshold between existence and non-existence of solutions which exist up to a maximal value λ*. This is referred in literature as the regularity issue for extremal solutions (see for instance [8, 20, 24, 33]).

Here we mention some recent papers on a semilinear elliptic system of a cooperative type which are closely related with our work. M. Montenegro in [31] studied elliptic systems of the form Δ⁢u=λ⁢f⁢(x,u,v) and Δ⁢v=μ⁢g⁢(x,u,v) defined in Ω, which is a smooth bounded domain under homogeneous Dirichlet boundary conditions. Under some suitable assumptions, which include in particular that the systems are cooperative, it was proved that there exists a monotone continuous curve Υ in the positive quadrant 𝒬 separating this set into two connected components: U “below” Υ, where there are C1⁢(Ω¯) minimal positive solutions, and V “above” Υ, where there is no such solution. For points on Υ there are weak solutions in the sense of the weighted Lebesgue space Ld1⁢(Ω), where d⁢(x) is the distance to the boundary ∂⁡Ω. Linearized stability of solutions in U is also proved. The existence proof uses sub- and supersolutions, and the existence of weak solutions is shown by a limiting argument involving a priori estimates in Ld1⁢(Ω) for classical solutions.

A question that attracted a lot of attention is the regularity of the extremal solution. For the scalar case (Pλ), F. Mignot and J.-P. Puel [29] studied regularity results to certain nonlinearities, namely, g⁢(u)=eu, g⁢(u)=um with m>1, g⁢(u)=1(1-u)k with k>0. Very recently, this analysis was complemented by N. Ghoussoub and Y. Guo [23] for the MEMS case in a bounded domain Ω under zero Dirichlet boundary condition, among other refined properties for stable steady states they proved that extremal solutions are smooth if 1≤N≤7 and N=8 is the critical dimension for this class of problems.

For elliptic systems, the stability inequality was first established in the study of Liouville theorems and De Giorgi’s conjecture for systems, see [21]. There is a correspondence between regularity of extremal solutions and Liouville theorems up to blow up analysis and scaling. This inequality was used to establish regularity results in [5] for systems and in [6] for the fourth-order case. C. Cowan [4] considered the particular case of nonlinearities of Gelfand type, that is, when f⁢(x,u,v)=ev and g⁢(x,u,v)=eu. He studied the regularity of the extremal solutions on the critical curve, precisely, he proved that if 3≤N≤9 and (N-2)8<μ*λ*<8(N-2) then the associated extremal solutions are smooth. This implies that N=10 is the critical dimension for Gelfand systems, because the scalar equation related with this class of problems may be singular if N≥10. Later, C. Cowan and M. Fazly in [5] examined the elliptic systems given by

and

with zero Dirichlet boundary condition in a bounded convex domain Ω. They proved that for a general nonlinearities f and g the extremal solutions associated with (1.1) are bounded when N≤3. For a radial domain, they proved the extremal solution are bounded provided that N<10. The extremal solutions associated with (1.2) are bounded in the case where f is a general nonlinearity and g⁢(v)=(1+v)q for 1<q<+∞ and N≤3. For the explicit nonlinearities of the form f⁢(u)=(1+u)p and g⁢(v)=(1+v)q certain regularity results are also obtained in higher dimensions for (1.1) and (1.2).

In the recent years, this class of problems has two natural fourth-order generalizations and extensions. D. Cassani, J. M. do Ó and N. Ghoussoub in [2] considered the problem of the form

with the biharmonic operator Δ2 and subject to Dirichlet conditions where η denotes the outward pointing unit normal to ∂⁡Ω. In the physical model, they consider the plate situation, in which the flexural rigidity is now allowed, whose effects, however, dominates over the stretching tension and neglecting non-local contributions. Since there is no maximum principle for Δ2 with Dirichlet boundary conditions for general domains, authors exploit the positivity of the Green function due to T. Boggio [1] and consider problem (1.3) restrict to the ball. After that, C. Cowan and N. Ghoussoub [6] studied the fourth-order problem of the form

with Navier boundary conditions where f is one of the following nonlinearities:

Note that one can regard the fourth-order equation (1.4) as a system of the following type:

Using this approach led them to prove regularity results for semi-stable solutions and hence for the extremal solutions using a stability inequality obtained for the elliptic system (1.5) associated with the problem (1.4).

1.2 Statement of Main Results

The main goal of this article is to provide a supplement for the ongoing studies of nonlinear eigenvalue problems of MEMS type, as this is the case for references [4, 5, 31]. Our first result deals with the existence of a curve that splits the positive quadrant into two connected components.

Theorem 1.2 and Theorem 1.3 contain upper and lower estimates for the critical curve. These estimates depend only on f,g,|Ω| and the dimension N, namely:

In the next two theorems we discuss the monotonicity properties of the critical curve for system (Sλ,μ). We mention that similar results have been proved for the scalar case (Pλ) in [19, 23]. In [19], it was shown that the permittivity profile g can change the bifurcation diagram and alter the critical dimension for compactness for the equation (Pλ).

Analogously to the scalar case (see [23]), we can define the notion of extremal solution of (Sλ,μ) for points on the critical curve. Precisely, let us consider a sequence (λn,μn) below the critical curve converging to a point (λ*,μ*) on the critical curve. In view of Theorem 1.1, we can consider the minimal solution (uλn,vμn) of system (Sλn,μn). Now, we can define the extremal solution (u*,v*) at (λ*,μ*) by passing to the limit when n→+∞, namely

The following theorem deals with regularity properties for solutions of (Sλ,μ). The main idea is to apply an appropriate test function in the stability inequality (see Lemma 3.5 below). This inequality is the main trick to tackle the problem for the case of systems and fourth-order equations. This kind of argument involving the stability inequality and Moser’s iteration method has been used by M. Crandall and P. Rabinowitz [7] and was originated in Harmonic maps and differential geometry.

1.3 Outline

This paper is organized as follows. In the next section we bring some auxiliary results used in the text. Moreover, we study the existence of a critical curve, extremal parameter and minimal solutions. We also establish upper and lower bounds for the critical curve Γ and monotonicity results for the extremal parameter. In Section 3 we obtain some estimates and properties for the branch of minimal solutions that allows us to prove the regularity result about the extremal solution.

2.1 Critical Curve

For each fixed θ>0 consider the line Lθ={λ>0:(λ,θ⁢λ)∈Λ}. Observe that Lemma 2.3 and Lemma 2.4 imply that for each fixed θ, the line Lθ is nonempty and bounded. This allows us to define the curve Γ:(0,+∞)→𝒬 by Γ⁢(θ):=(λ*⁢(θ),μ*⁢(θ)) where λ*⁢(θ):=sup⁡Lθ and μ*⁢(θ)=θ⁢λ*⁢(θ).

2.2 Upper and Lower Bounds for the Critical Curve

According to N. Ghoussoub and Y. Guo [23], the lower bound for the critical parameter is useful to prove existence of solutions for (Pλ). The following lemma will be the main tool to obtain the estimates contained in Theorem 1.2 and gives more computationally accessible lower estimates for the critical curve.

2.3 Monotonicity Results for the Extremal Parameter

Let GΩ⁢(x,ξ)=G⁢(x,ξ) be the Green’s function of the Laplace operator for the region Ω, with G⁢(x,ξ)=0 if x∈∂⁡Ω. We shall write (un,Ω⁢(x),vn,Ω⁢(x))=(un⁢(x),vn⁢(x)) for the sequence obtained by the interaction process as follows: (u0,v0)=(0,0) in Ω and

It is easy to see that the sequence above converges uniformly for a minimal solution of (Sλ,μ) provided that 0<λ<λ∗ and 0<μ<Γ⁢(λ). This construction will help us to prove the monotonicity result for λ* stated in Theorem 1.4.

We shall use the Schwarz symmetrization method. Let BR=BR⁢(0) be the Euclidean ball in ℝN with radius R>0 centred at origin such that |BR|=|Ω|, and let u♯ be the symmetrization of u, then it is well known that u♯ depends only on |x| and u♯ is a decreasing function of |x|.