On Lane-Emden systems with singular nonlinearities and applications to MEMS
aa r X i v : . [ m a t h . A P ] J a n ON LANE-EMDEN SYSTEMS WITH SINGULAR NONLINEARITIES ANDAPPLICATIONS TO MEMS
JO ˜AO MARCOS DO ´O AND RODRIGO G. CLEMENTE
Abstract.
In this paper we analyse the Lane-Emden system − ∆ u = λf ( x )(1 − v ) in Ω − ∆ v = µg ( x )(1 − u ) in Ω0 ≤ u, v < u = v = 0 on ∂ Ω ( S λ,µ )where λ and µ are positive parameters and Ω is a smooth bounded domain of R N ( N ≥ λ, µ )-plane into twodisjoint sets O and O such that the problem ( S λ,µ ) has a smooth minimal stable solution ( u λ , v µ ) in O , while for ( λ, µ ) ∈ O there are no solutions of any kind. We also establish upper and lower estimatesfor the critical curve Γ and regularity results on this curve if N ≤
7. Our proof is based on a delicatecombination involving maximum principle and L p estimates for semi-stable solutions of ( S λ,µ ). Introduction
In this paper we deal with Hamiltonian systems of coupled singular elliptic equations of second-orderof the form − ∆ u = λf ( x )(1 − v ) in Ω , − ∆ v = µg ( x )(1 − u ) in Ω , ≤ u, v < ,u = v = 0 on ∂ Ω , ( S λ,µ )where λ and µ are positive parameters, Ω is a smooth bounded domain of R N ( N ≥
2) and f and g satisfythe following conditions: f, g ∈ C α (Ω) for some α ∈ (0 , ≤ f, g ≤ f, g > H )1.1. Motivation and related results.
System ( S λ,µ ) can be seen as a Lane-Emden type systemwith nonlinearities with negative exponents [18, 28, 34, 35]. A lot of work has been devoted to existenceand nonexistence of solutions to elliptic systems with continuous power like nonlinearities, among whichwe recall [11–15, 26, 30] and the survey [10], just recently Lane-Emden type singular nonlinearities havebeen considered in [22]. Here we address the problem of studying existence, non-existence and regularityresults by means of the nonlinear eigenvalue problem ( S λ,µ ), in which for the sake of clarity we consider Mathematics Subject Classification.
Key words and phrases.
Nonlinear PDE of elliptic type, singular nonlinearity, elliptic systems, semi-stable solution,extremal solution, regularity of extremal solutions.Research partially supported by the National Institute of Science and Technology of Mathematics INCT-Mat, CAPESand CNPq. a Coulomb nonlinear source though most results extend to more general situations. Related results forsystems with continuous nonlinearities have been obtained in [25, 31].Another important motivation to consider ( S λ,µ ) comes from recent works on the study of the equationsthat models MEMS − ∆ v = λ g ( x )(1 − v ) in Ω , ≤ v < ,v = 0 on ∂ Ω . ( P λ )Micro-electromechanical systems (MEMS) are often used to combine electronics with micro size mechanicaldevices in the design of various types of microscopic machinery. MEMS devices have therefore become keycomponents of many commercial systems, including accelerometers for airbag deployment in vehicles, inkjet printer heads, optical switches and chemical sensors.Nonlinear interaction described in terms of coupling of semilinear elliptic equations has revealed throughthe last decades a fundamental tool in studying nonlinear phenomena, see e.g. [3,9,13,14,16] and referencestherein. In all the above contexts the nonlinearity is fairly represented by a continuous function. Morerecently, a rigorous mathematical approach in modeling and designing Micro Electro Mechanical Systemshas demanded the need to consider also nonlinearities which develop singularities. In a nutshell, one maythink of MEMS’ actuation as governed by the dynamic of a micro plate which deflects towards a fixedplate, 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 ellipticPDE ( P λ ), where Ω is a bounded smooth domain in R N and the positive function g is bounded and relatedto 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 ( P λ ) is retained by the discontinuity of the nonlinearity which blowsup as v → − and this corresponds in applications to a snap through of the device.The general goal on the study of ( P λ ) is to analyse the structure of the branch of solutions as wellas their qualitative properties. The role of the positive parameter λ is that of tuning the drop voltage,whence from the PDE point of view, yields the threshold between existence and non-existence of solutionswhich exist up to a maximal value λ ∗ . This is referred in literature as the regularity issue for extremalsolutions, see for instance [8, 20, 24, 33].Here we mention some recent papers on semilinear elliptic system of cooperative type which are closelyrelated with our work. M. Montenegro in [31] studied elliptic systems of the form ∆ u = λf ( x, u, v ),∆ v = µg ( x, u, v ) defined in Ω a smooth bounded domain under homogeneous Dirichlet boundaryconditions. 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 thisset into two connected components: U “below” Υ, where there are C (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 theweighted Lebesgue space L d (Ω) , where d ( x ) is the distance to the boundary ∂ Ω. Linearized stability ofsolutions in U is also proved. The existence proof uses sub- and supersolutions, and the existence of weaksolutions is shown by a limiting argument involving a priori estimates in L d (Ω) for classical solutions.A question that attracted a lot of attention is the regularity of the extremal solution. For the scalar case( P λ ), F. Mignot and J-P. Puel [29] studied regularity results to certain nonlinearities, namely, g ( u ) = e u , g ( u ) = u m with m > g ( u ) = 1 / (1 − u ) k with k >
0. Very recently, this analysis was complemented byN. Ghoussoub and Y. Guo [23] for the MEMS case in a bounded domain Ω under zero Dirichlet boundarycondition, among other refined properties for stable steady states they proved that extremal solutions aresmooth if 1 ≤ N ≤ 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 andDe Giorgi’s conjecture for systems, see [21]. There is a correspondence between regularity of extremalsolutions and Liouville theorems up to blow up analysis and scaling. This inequality was used to establishregularity results in [5] for systems and [6] for the fourth order case. C. Cowan [4] considered the particular N LANE-EMDEN SYSTEMS WITH SINGULAR NONLINEARITIES 3 case of nonlinearities of Gelfand type, that is, when f ( x, u, v ) = e v and g ( x, u, v ) = e u . He studied theregularity of the extremal solutions on the critical curve, precisely, he proved that if 3 ≤ N ≤ N − / < µ ∗ /λ ∗ < / ( N −
2) then the associated extremal solutions are smooth. This implies that N = 10 is the critical dimension for the Gelfand systems, because the scalar equation related with thisclass of problems may be singular if N ≥
10. Later, C. Cowan and M. Fazly in [5] examined the ellipticsystems given by − ∆ u = λf ′ ( u ) g ( v ) , − ∆ v = µf ( u ) g ′ ( v ) in Ω , (1.1)and − ∆ u = λf ( u ) g ′ ( v ) , − ∆ v = µf ′ ( u ) g ( v ) in Ω (1.2)with zero Dirichlet boundary condition in a bounded convex domain Ω. They proved that for a generalnonlinearities f and g the extremal solution associated with (1.1) are bounded when N ≤
3. For a radialdomain, they proved the extremal solution are bounded provided that
N <
10. The extremal solutionassociated with (1.2) are bounded in the case where f is a general nonlinearity and g ( v ) = (1 + v ) q for1 < 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 ´O and N. Ghoussoub in [2] considered the problem ∆ u = λf ( x )(1 − v ) in Ω , ≤ u < ,u = ∂u∂η = 0 on ∂ Ω (1.3)with the biharmonic operator ∆ and subject to Dirichlet conditions where η denotes the outward pointingunit normal to ∂ Ω. In the physical model, they consider the plate situation in which flexural rigidity is nowallowed whose effects however dominates over the stretching tension, neglecting non-local contributions.Since there is no maximum principle for ∆ with Dirichlet boundary conditions for general domains, theauthors exploit the positivity of the Green function due to T. Boggio [1] and consider problem (1.3) restrictto the ball. After that, C. Cowan and N. Ghoussoub [6] studied the fourth order problem ∆ u = λf ( u ) in Ω , ≤ u < ,u = ∆ u = 0 on ∂ Ω (1.4)with Navier boundary conditions where f is one of following nonlinearities: f ( u ) = e u , f ( u ) = (1 + u ) p or f ( u ) = (1 − u ) − p . Note that one can view the fourth order equation (1.4) as a system of the followingtype − ∆ v = λf ( u ) in Ω , − ∆ u = v on ∂ Ω ,u = v = 0 on ∂ Ω . (1.5)Using this approach, they proved regularity results for semi-stable solutions and hence for the extremalsolutions 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 theongoing 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 split the positive quadrant into two connectedcomponents.
J.M. DO ´O AND R.G. CLEMENTE
Theorem 1.1.
Suppose that condition ( H ) holds. Then, there exists a curve Γ that separates the positivequadrant Q of the ( λ, µ ) -plane into two connected components O and O . For ( λ, µ ) ∈ O , problem ( S λ,µ ) has a positive classical minimal solution ( u λ , v λ ) . Otherwise, if ( λ, µ ) ∈ O , there are no solutions. The Theorem 1.2 and Theorem 1.3 contain upper and lower estimates for the critical curve. Theseestimates depend only on f, g, | Ω | and the dimension N , namely, Theorem 1.2.
Suppose f, g satisfy ( H ) . Then the region O is nonempty, more precisely, there exist apositive constant C N which depends only of the dimension N such that (0 , a ( f, | Ω | ,N ) ] × (0 , a ( g, | Ω | ,N ) ] ⊂ O , where a ( f, | Ω | ,N ) := C N Ω f ( x ) (cid:18) ω N | Ω | (cid:19) /N , a ( g,R,N ) := C N Ω g ( x ) (cid:18) ω N | Ω | (cid:19) /N and C N = max (cid:26) N , N − (cid:27) . Theorem 1.3.
Suppose f, g satisfy ( H ) . Assume that inf Ω f ( x ) > (respectively inf Ω g ( x ) > ), then λ ∗ ≤ µ
27 1inf Ω f ( x ) (cid:18) respectively µ ∗ ≤ µ
27 1inf Ω g ( x ) (cid:19) , where µ is the first eigenvalue of ( − ∆ , H (Ω)) . Therefore, if inf Ω f ( x ) > and inf Ω g ( x ) > the region O is bounded, precisely, O ⊂ (cid:18) , µ
27 1inf Ω f ( x ) (cid:19) × (cid:18) , µ
27 1inf Ω g ( x ) (cid:19) . In the next two theorems we discuss the monotonicity properties of the critical curve for system( S λ,µ ). We mention that similar results have been proved for the scalar case ( P λ ) in [19, 23]. In [19], itwas shown that the permittivity profile g can be change the bifurcation diagram and alter the criticaldimension for compactness for the equation ( P λ ). Theorem 1.4.
Suppose that condition ( H ) holds. If ( S λ,µ ) has a solution in Ω , then it also has a solutionfor any subdomain Ω ′ ⊂ Ω for which the Green’s function exists. Furthermore, λ ∗ (Ω ′ ) ≥ λ ∗ (Ω) and forthe corresponding minimal solutions, we have u Ω ′ ( x ) ≤ u Ω ( x ) and v Ω ′ ( x )) ≤ v Ω ( x ) for all x ∈ Ω . Theorem 1.5.
Let f, g satisfying ( H ) and f ♯ , g ♯ the Schwarz symmetrization of f and g respectively.Then λ ∗ (Ω , f, g ) ≥ λ ∗ ( B R , f ♯ , g ♯ ) and for each λ ∈ (0 , λ ∗ ( B R , f, g )) we have Γ (Ω ,f,g ) ( λ ) ≥ Γ ( B R ,f ♯ ,g ♯ ) ( λ ) . Analogously to scalar case (see [23]), we can define the notion of extremal solution of ( S λ,µ ) forpoints on the critical curve. Precisely, let us consider a sequence ( λ n , µ n ) below the critical curve convergingto a point ( λ ∗ , µ ∗ ) on the critical curve. In view of Theorem 1.1, we can consider the minimal solution( u λ n , v µ n ) of System ( S λ n ,µ n ). Now, we can define the extremal solution ( u ∗ , v ∗ ) at ( λ ∗ , µ ∗ ) by passing tothe limit when n → + ∞ , namely, ( u ∗ , v ∗ ) = lim n → + ∞ ( u λ n , v µ n ) . The following theorem deals with regularity properties for solutions of ( S λ,µ ). The main idea is toapply an appropriate test function in the stability inequality (see Lemma 3.2 below). This inequality isthe main trick to tackle the problem for the case of systems and fourth order equations. This kind ofargument involving stability inequality and Moser’s iteration method has been used by M. Crandall andP. Rabinowitz [7] and was originated in Harmonic maps and differential geometry. Theorem 1.6.
Assume that f, g = 1 . Then the extremal solution ( u ∗ , v ∗ ) of System ( S λ ∗ ,µ ∗ ) is smoothwhen N ≤ . N LANE-EMDEN SYSTEMS WITH SINGULAR NONLINEARITIES 5
Remark 1.1.
Observe that Theorem 1.6 determines the critical dimension for this class of Lane-Emdensystems, precisely determine the dimension N ∗ such that the extremal solution is smooth when N < N ∗ and singular when N ≥ N ∗ . Indeed, if we consider Ω to be the unit ball, u = v and λ = µ , then the systemturns into a scalar equation and the optimal results are known. For instance, the function u ∗ ( x ) = 1 −| x | / is a singular solution for − ∆ u = λ/ (1 − u ) if N ≥ (see [23, Theorem 1.3]). Outline.
This paper is organized as follows. In the next section we bring some auxiliary resultsused in the text. Moreover, we study the existence of a critical curve, extremal parameter and minimalsolutions. We also establish upper and lower bounds for the critical curve Γ and monotonicity results forthe extremal parameter. In Section 3 we obtain some estimates and properties for the branch of minimalsolutions that allow us to prove the regularity result about the extremal solution.2.
A critical curve: existence of classical solutions
The main goal of this section is to prove Theorems 1.1, 1.2 and 1.3. Precisely, by the method ofsub-super solutions we prove that there exists a non-increasing continuous function Γ of the parameter λ such that ( S λ,µ ) has at least one solution for 0 < µ < Γ( λ ) whereas ( S λ,µ ) has no solutions for µ > Γ( λ ).In what follows unless otherwise stated, by solution we mean a classical solution of class C (Ω). For thesake of completeness, we briefly sketch the proofs of the next lemmas. For more details we refer thereader [4, 5, 31] Lemma 2.1.
Let λ and µ positive parameters such that there exists a classical super solution ( U, V ) for ( S λ,µ ) . Then there exists a classical solution ( u, v ) of ( S λ,µ ) such that u ≤ U and v ≤ V .Proof. Setting ( u , v ) = ( U, V ) we can define ( u n , v n ) inductively as follows − ∆ u n = λf ( x )(1 − v n − ) in Ω , − ∆ v n = µg ( x )(1 − u n − ) in Ω , ≤ u n , v n < ,u n = v n = 0 on ∂ Ω . By the maximum principle, we have 0 < u n ≤ u n − ≤ . . . u ≤ u and 0 < v n ≤ v n − ≤ . . . v ≤ u . Thus,there exists ( u, v ) such that 0 ≤ u = lim n →∞ u n ≤ U < ≤ v = lim n →∞ v n ≤ V < C ,α (Ω) to a solution ( u, v )of ( S λ,µ ) and in particular different from zero. (cid:3) We now state and prove a monotonicity result on the coordinates of a solution of ( S λ,µ ), precisely, Lemma 2.2.
Suppose that ( u, v ) is a smooth solution of ( S λ,µ ) where < µ ≤ λ. Then µu/λ ≤ v ≤ u a.ein Ω .Proof. Take the difference of the equations in ( S λ,µ ), multiplying this equation by ( u − v ) − and integratingby parts we have Z Ω |∇ ( u − v ) − | dx = Z Ω (cid:18) λ (1 − v ) − µ (1 − u ) (cid:19) ( u − v ) − dx. Since the right hand side is nonpositive and the left hand side is nonnegative, we see that ( u − v ) − = 0a.e. in Ω and so u ≥ v a.e. in Ω. Now, since u ≥ v , − ∆ (cid:16) v − µλ u (cid:17) = µ (cid:18) − u ) − − v ) (cid:19) ≥ . Thus, µλ u ≤ v and we finish the proof. (cid:3) J.M. DO ´O AND R.G. CLEMENTE
We are going to prove that ( S λ,µ ) has a classical solution for λ and µ sufficiently small, moreprecisely, the set Λ := { ( λ, µ ) ∈ Q : ( S λ,µ ) has a classical solution } has nonempty interior. Lemma 2.3.
There exists λ > such that (0 , λ ] × (0 , λ ] ⊂ Λ .Proof. Let B R be a ball of radius R such that Ω ⊂ B R and let µ ,R be the first eigenvalue of the Dirichletboundary value problem ( − ∆ , H (Ω)) and denote the corresponding eigenfunction by ψ ,R which we mayassume to be positive and also that sup B R ψ ,R = 1. Now we show that there exists θ > ψ = θψ ,R is a super-solution to ( S λ,λ ) provided λ > θ ∈ (0 ,
1) such that 0 < − θψ ,R < B . Thus − ∆ ψ = µ ,R θψ ,R ≥ λf ( x )(1 − ψ ) = λf ( x )(1 − θψ ,R ) in Ω , − ∆ ψ = µ ,R θψ ,R ≥ λg ( x )(1 − ψ ) = λg ( x )(1 − θψ ,R ) in Ω , provided µ ,R θψ ,R (1 − θψ ,R ) ≥ λ max { f ( x ) , g ( x ) } . Notice that s := inf x ∈ Ω θψ ,R < s :=sup x ∈ Ω θψ ,R <
1, and s , s depend of R . Setting Z ( s ) := s (1 − s ) , it is easy to see that we canchoose λ > µ ,R inf x ∈ Ω Z ( θψ ,R ( x )) ≥ λ max { sup Ω g ( x ) , sup Ω f ( x ) } . Thus,using Lemma 2.1 we conclude that ( λ, µ ) ∈ Λ, for all λ, µ ∈ (0 , λ ]. (cid:3) Lemma 2.4. Λ is bounded.Proof. Let ( λ, µ ) ∈ Λ and ( u, v ) the corresponding solution of ( S λ,µ ). Multiplying the first equation in( S λ,µ ) by ψ ,R and integrating by parts implies that | B R | µ ,R ≥ λ Z B R f ( x ) ψ ,R d x. Analogously, multiplying the second equation in ( S λ,µ ) by ψ ,R we obtain | B R | µ ,R ≥ µ Z B R g ( x ) ψ ,R d x and therefore Λ is bounded. (cid:3) Now we state that Λ is a convex set, precisely,
Lemma 2.5. If ( λ ′ , µ ′ ) ∈ Q and λ ′ ≤ λ and µ ′ ≤ µ for some ( λ, µ ) ∈ Λ then ( λ ′ , µ ′ ) ∈ Λ .Proof. It follows from Lemma 2.1. Indeed, the solution associated to the pair ( λ, µ ) ∈ Λ turns out to be asuper-solution to ( S λ ′ ,µ ′ ). (cid:3) Critical curve.
For each fixed θ > L θ = { λ > λ, θλ ) ∈ Λ } . Observethat Lemma 2.3 and Lemma 2.4 implies that for each θ fixed, the line L θ is nonempty and bounded.This allow us to define the curve Γ : (0 , + ∞ ) → Q by Γ( θ ) := ( λ ∗ ( θ ) , µ ∗ ( θ )) where λ ∗ ( θ ) := sup L θ and µ ∗ ( θ ) = θλ ∗ ( θ ). Proof of Theorem 1.1.
Define O = Λ \ Γ. Given ( λ , µ ) , ( λ , µ ) ∈ O , there exist θ , θ > µ = θ λ and µ = θ λ . We can define, using the Lemma 2.5, a path linking ( λ , µ ) to (0 ,
0) andanother path linking (0 ,
0) to ( λ , µ ). Follows that O is connected. The Lemma 2.1 implies that foreach ( λ, µ ) ∈ O there exists a positive minimal classical solution ( u λ , v µ ) for problem ( S λ,µ ). Now, define O = Q \ { Λ ∪ Γ } . Let ( λ , µ ) , ( λ , µ ) ∈ O . Take ( λ max , µ max ) ∈ O , where λ max = max { λ , λ } and µ max = max { µ , µ } . We can take a path linking ( λ , µ ) to ( λ max , µ max ) and another path linking( λ max , µ max ) to ( λ , µ ). Follows that O is connected. (cid:3) N LANE-EMDEN SYSTEMS WITH SINGULAR NONLINEARITIES 7
Upper and lower bounds for the critical curve.
As noticed by N. Ghoussoub and Y. Guo[23], the lower bound for the critical parameter is useful to prove existence of solutions for ( P λ ). Thefollowing lemma will be the main tool to obtain the estimates contained in Theorem 1.2 and gives morecomputationally accessible lower estimates for the critical curve. Lemma 2.6.
Assume that
Ω = B = B R and f, g are radial, that is, f ( x ) = f ( | x | ) and g ( x ) = g ( | x | ) , forall x ∈ B . Then (0 , a ( f,R,N ) ] × (0 , a ( g,R,N ) ] ⊂ Λ where a ( f,R,N ) := C N B f ( x ) 1 R , a ( g,R,N ) := C N B g ( x ) 1 R . and C N = max (cid:26) N , N − (cid:27) . Proof.
Notice that the function w ( x ) := 1 / (cid:0) − | x | /R (cid:1) satisfies − ∆ w = 2 N R ≥ N R sup B f f ( x ) h − (cid:16) − | x | R (cid:17)i = 8 N R sup B f f ( x )(1 − w ) . Similarly, − ∆ w ≥ N R sup B g g ( x )(1 − w ) . Thus, for λ ≤ N/ (27 R sup B f ) and µ ≤ N/ (27 R sup B g ) we have that ( w, w ) is a super-solution of( S λ,µ ) in B . Similarly, we can see that, taking v ( x ) := 1 − ( | x | /R ) / , the pair ( v, v ) is a super-solution for( S λ,µ ) in B provided that λ ≤ (6 N − / (9 R sup B f ) and µ ≤ (6 N − / (9 R sup B g ), which completesthe proof. (cid:3) Proposition 2.1.
Assume that
Ω = B = B R and f ( x ) = | x | α , g ( x ) = | x | β with α, β ≥ , then (0 , a ( α,R,N ) ] × (0 , b ( β,R,N ) ] ⊂ Λ , where a ( α,R,N ) := max (cid:26) α )( N + α )27 , (2 + α )(3 N + α − (cid:27) R α and b ( β,R,N ) := max (cid:26) β )( N + β )27 , (2 + β )(3 N + β − (cid:27) R β . Proof.
Consider the function w ( α,R ) ( x ) = 1 / (cid:0) − | x | α /R α (cid:1) . Using a similar computation as we havedone in the previous lemma we can prove that the pair ( w ( α,R ) , w ( β,R ) ) is a super-solution of ( S λ,µ ) in B provided that λ ≤ α )( N + α )27 R α and µ ≤ β )( N + β )27 R β . The same holds for the function w ( x ) = 1 − ( | x | /R ) (2+ α ) / if λ ≤ (2 + α )(3 N + α − R α and µ ≤ (2 + β )(3 N + β − R β . (cid:3) J.M. DO ´O AND R.G. CLEMENTE
Proof of Theorem 1.3.
Consider ( λ, µ ) ∈ Λ and ( u, v ) the corresponding solution of ( S λ,µ ). Let µ anddenote the corresponding positive eigenfunction by ψ . Taking ψ as a test function in the first equationof ( S λ,µ ) and using integration by parts we obtain Z Ω (cid:18) − µ u + λf ( x )(1 − v ) (cid:19) ψ d x = 0which implies that λ > λ ∗ when − µ u + λf ( x )(1 − v ) > . (2.1)After a simple calculation we find that (2.1) holds when λ > µ
27 1inf Ω f ( x ) . Using the same approach in the second equation we finish the proof. (cid:3)
Monotonicity results for the extremal parameter.
Let G Ω ( x, ξ ) = G ( x, ξ ) be the Green’sfunction of the Laplace operator for the region Ω, with G ( x, ξ ) = 0 if x ∈ ∂ Ω. We shall write( u n, Ω ( x ) , v n, Ω ( x )) = ( u n ( x ) , v n ( x )) for the sequence obtained by the interaction process as follows:( u , v ) = (0 ,
0) in Ω and u n ( x ) = Z Ω λf ( x ) G ( x, ξ )(1 − v n − ) dξ in Ω ,v n ( x ) = Z Ω µg ( x ) G ( x, ξ )(1 − u n − ) dξ in Ω . (2.2)It is easy to see that the sequence above converges uniformly for a minimal solution of ( S λ,µ ) providedthat 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. Proof of Theorem 1.4.
Let ( u n, Ω ′ , v n, Ω ′ ) be defined as in (2.2) with Ω replaced by Ω ′ . Using thecorresponding Green’s functions for the subdomains Ω ′ ⊂ Ω satisfy the inequality G Ω ′ ( x, ξ ) ≤ G Ω ( x, ξ ) wehave u , Ω ′ ( x ) = Z Ω ′ λf ( x ) G Ω ′ ( x, ξ ) dξ ≤ Z Ω λf ( x ) G Ω ( x, ξ ) dξ in Ω ′ ,v , Ω ′ ( x ) = Z Ω ′ µg ( x ) G Ω ′ ( x, ξ ) dξ ≤ Z Ω µg ( x ) G Ω ( x, ξ ) dξ in Ω ′ . By induction we conclude that u n, Ω ′ ( x ) ≤ u n, Ω ( x ) and v n, Ω ′ ( x ) ≤ v n, Ω ( x ) in Ω ′ . On the other hand,since u n, Ω ( x ) ≤ u n +1 , Ω ( x ) and v n, Ω ( x ) ≤ v n +1 , Ω ( x ) in Ω, for each n , we get that u n, Ω ′ ( x ) ≤ u Ω ( x ) and v n, Ω ′ ( x ) ≤ v Ω ( x ) in Ω ′ and we are done. (cid:3) Corollary 2.1.
Suppose f , f , g , g : Ω → R satisfy condition ( H ) and f ( x ) ≤ f ( x ) and g ( x ) ≤ g ( x ) for all x ∈ Ω , then λ ∗ ( f , g ) ≥ λ ∗ ( f , g ) and for each λ ∈ (0 , λ ∗ ( f , g )) . Furthermore u ( x ) ≤ u ( x ) and v ( x ) ≤ v ( x ) for all x ∈ Ω for the corresponding minimal solutions. If f ( x ) < f ( x ) or g ( x ) < g ( x ) ona subset of positive measure, then u ( x ) < ( u ( x ) and v ( x ) < v ( x ) for all x ∈ Ω . We shall use Schwarz symmetrization method. Let B R = B R (0) the Euclidean ball in R N withradius R > | B R | = | Ω | , and let u ♯ be the symmetrization of u , then it iswell-known that u ♯ depends only on | x | and u ♯ is a decreasing function of | x | . Proof of Theorem 1.5.
For each λ ∈ (0 , λ ∗ ( B R , f, g )) and µ ∈ (0 , Γ ( B R ,f ♯ ,g ♯ ) ( λ )) we consider the minimalsequence ( u n , v n ) for ( S λ,µ ) as defined in (3.1), and let ( b u n , b v n ) be the minimal sequence for the N LANE-EMDEN SYSTEMS WITH SINGULAR NONLINEARITIES 9 corresponding Schwarz symmetrized problem: − ∆ u = λf ♯ ( x )(1 − v ) in B R , − ∆ v = µg ♯ ( x )(1 − u ) in B R ,
Since sup B R f ♯ = sup Ω f and sup B R g ♯ = sup Ω g , setting R = ( | Ω | /ω N ) /N theproof follows as an applications of Theorem 1.5 and Lemma 2.6. (cid:3) The branch of minimal solutions
Next, assuming the existence of solutions for System ( S λ,µ ), we obtain also existence and uniquenessof minimal solution. Lemma 3.1.
For any < λ < λ ∗ and < µ < Γ( λ ) , there exists a unique minimal solution ( u, v ) of( S λ,µ ).Proof. This minimal solution is obtained as the limit of the sequence of pair of functions ( u n , v n )constructed recursively as follows: ( u , v ) = (0 ,
0) in Ω and for each n = 1 , , . . . , ( u n , v n ) is the uniquesolution of the boundary value problem: − ∆ u n = λf ( x )(1 − v n − ) in Ω , − ∆ v n = µg ( x )(1 − u n − ) in Ω , u ≡ ≥ V > v ≡ U ≥ u n − and V ≥ v n − in Ω. Thus, − ∆( U − u n ) = λf ( x ) (cid:20) − V ) − − u n − ) (cid:21) ≥ , − ∆( V − v n ) = µg ( x ) (cid:20) − U ) − − v n − ) (cid:21) ≥ ,U − u n = V − v n = 0 on ∂ Ω . By the maximum principle we conclude that 1 > U ≥ u n > > V ≥ v n > u n , v n ) is a monotone increasing sequence. Therefore, ( u n , v n )converges uniformly to a solution ( u, v ) of ( S λ,µ ), which by construction is unique in this class of minimalsolutions. (cid:3) We can introduce for any solution u of ( P λ ), the linearized operator at u defined by L u,λ = − ∆ − λf ( x )(1 − u ) and its eigenvalues { µ k,λ ( u ); k = 1 , , ... } . The first eigenvalue is then simple and canbe characterized variationally by µ ,λ ( u ) = inf (cid:26) h L u,λ φ, φ i H (Ω) ; φ ∈ C ∞ (Ω) , Z Ω | φ ( x ) | dx = 1 (cid:27) . Stable solutions (resp., semi-stable solutions ) of ( S ) λ are those solutions u such that µ ,λ ( u ) > µ ,λ ( u ) ≥ ≤ N ≤ λ close enough to λ ∗ that there exists a unique second branch of solutions for ( P λ )bifurcating from u ∗ .In the case that ( u, v ) is a solution of ( S λ,µ ) we consider the first eigenvalue ν = ν (( λ, µ ) , ( u, v )) ofthe linearization L := −−→ ∆ − A ( x ) around ( u, v ) under Dirichlet boundary conditions, where −→ ∆Φ = (cid:18) ∆ φ ∆ φ (cid:19) and A ( x ) := (cid:18) a ( x ) a ( x ) 0 (cid:19) = λf ( x )(1 − v ( x )) µg ( x )(1 − u ( x )) ! that is, the eigenvalue problem L Φ = ν Φ , Φ ∈ W , (Ω) × W , (Ω) , namely, ν is the first eigenvalue of the problem − ∆ φ − λf ( x )(1 − v ) φ = νφ in Ω , − ∆ φ − µg ( x )(1 − u ) φ = νφ in Ω ,φ = φ = 0 on ∂ Ω . ( E ( λ,µ ) )We recall that in [36, Proposition 3.1] was proved that there exists a unique eigenvalue ν with strictlypositive eigenfunction φ = ( φ , φ ) of ( E ( λ,µ ) ), that is, φ i > i = 1 , Remark 3.1.
The first eigenvalue of the linearized single equation has a variational characterization; nosuch analogous formulation is available for our system.
Definition 3.1 (Stable and Semi-stable Solution) . A solution of ( S λ,µ ) is said to be stable (resp. semi-stable) if ν > (resp., ν ≥ ). Proposition 3.1.
Suppose that ( λ, µ ) ∈ Λ with < µ ≤ λ and we let ( u, v ) denote the minimal solutionof ( S λ,µ ) . Let φ , φ as in ( E ( λ,µ ) ) . Then φ φ ≥ µλ in Ω . Proof.
Take the difference equation in ( E ( λ,µ ) ) and use Lemma 2.2 to obtain − ∆( φ − φ ) − ν ( φ − φ ) + µ ( φ − φ )(1 − v ) ≥ ( µ − λ ) φ (1 − v ) in Ω . N LANE-EMDEN SYSTEMS WITH SINGULAR NONLINEARITIES 11
Now, define a elliptic operator L := − ∆ − ν . We have that L (cid:18) ψ − ψ + λ − µλ ψ (cid:19) + µ (1 − v ) (cid:18) ψ − ψ + λ − µλ (cid:19) ≥ L (cid:18) ψ − ψ + λ − µλ ψ (cid:19) + µ (1 − v ) ( ψ − ψ ) ≥ ( µ − λ ) φ (1 − v ) + λ − µλ L ( φ ) = 0Using the maximum principle, we have φ − φ + ( λ − µ ) φ /λ ≥ φ /φ ≥ µ/λ and this finish the proof. (cid:3) Estimates for minimal solutions.
The next result is crucial in our argument to obtain theregularity of semistable solutions of ( S λ,µ ). For the proof we refer the reader to [17, Lemma 3]. Lemma 3.2.
Let N ≥ and let ( u, v ) ∈ C (cid:0) Ω (cid:1) × C (cid:0) Ω (cid:1) denote a stable solution of − ∆ u = g ( v ) in Ω , − ∆ v = f ( u ) in Ω ,u = v = 0 on ∂ Ω , where f and g denote two nondecreasing C functions. Then for all ϕ ∈ C c (Ω) there holds Z Ω p f ′ ( u ) g ′ ( v ) ϕ d x ≤ Z Ω |∇ ϕ | d x. Now we follow the approach due to L. Dupaigne, A. Farina and B. Sirakov [17] adapted to MEMScase. The main idea is apply H¨older’s inequality and iterate both equations in System ( S λ,µ ). This methodis the key to obtain the optimal dimension for the regularity of extremal solutions. Proof of Theorem 1.6.
Let α >
1, multiply the equation − ∆ u = λ/ (1 − v ) by (1 − u ) − α − λ Z Ω (1 − v ) − [(1 − u ) − α −
1] d x = α Z Ω (1 − u ) − α − |∇ u | d x = 4 α ( α − Z Ω (cid:12)(cid:12)(cid:12) ∇ (cid:16) (1 − u ) − α + (cid:17)(cid:12)(cid:12)(cid:12) d x. We can test (1 − u ) − α/ / − p λµ Z Ω (1 − u ) − (1 − v ) − [(1 − u ) − α + − d x ≤ Z Ω (cid:12)(cid:12)(cid:12) ∇ (cid:16) (1 − u ) − α + (cid:17)(cid:12)(cid:12)(cid:12) d x. Combining these two previous inequalities and developing the square follows p λµ Z Ω (1 − u ) − α − (1 − v ) − ≤ λ ( α − α Z Ω (1 − u ) − α (1 − v ) − + 2 p λµ Z Ω (1 − u ) − α − (1 − v ) − (3.2)Denote X = Z Ω (1 − u ) − α − (1 − v ) − and Y = Z Ω (1 − u ) − α − (1 − v ) − α − . Now we need estimate the terms on the right-hand side. Take p = α/ ( α −
1) and q = α and using H¨olderinequality with this exponents we obtain Z Ω (1 − u ) − α (1 − v ) − ≤ X α − α Y α . (3.3) Given ǫ >
0, we can use Young’s inequality and Lemma 2.2 to obtain Z Ω (1 − u ) − α − (1 − v ) − ≤ ǫ √ λ √ µ Z Ω (1 − u ) − α (1 − v ) − + √ µ √ λ | Ω | ǫ . (3.4)Thus, by (3.2),(3.3) and (3.4) we obtain p λµX ≤ λ " ( α − α + ǫ X α − α Y α + | Ω | ǫ . By symmetry, we also have p λµY ≤ µ " ( α − α + ǫ Y α − α X α + | Ω | ǫ . Multiplying this equations we have − ( α − α + ǫ ! XY ≤ " ( α − α + ǫ | Ω | ǫ h X α − α Y α + X α Y α − α i + | Ω | ǫ . Choose ǫ = 1 /
16 and thus we can verify that for every 1 < α < ,
62, either X or Y must be bounded. Wecan suppose λ ≤ µ and by Lemma 2.2 we have u ≤ v . Thus follows that (1 − u ) − must be bounded, eitherin L p for p < ( α + 2) / L q for q < α + 5 /
3. We note that the second case does not occur, becauseotherwise the semistable solutions should be regular for dimension N ≤
22, but we already known that,in the scalar case, u ∗ ( x ) = 1 − | x | / is a singular solution when Ω is the unit ball and N ≥
8. Therefore,the first case must occur and consequently u ∗ is smooth for N ≤ (cid:3) Remark 3.2.
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E-mail address : [email protected] (R. Clemente) Department of Mathematics, Rural Federal University of Pernambuco52171-900, Recife, Pernambuco, Brazil
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