Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the p-Laplacian and application A. Aghajani
a,b∗
, A.M. Tehrani
a
a School
arXiv:1609.05437v1 [math.AP] 18 Sep 2016
b School
of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844-13114, Iran. of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box: 19395-5746, Tehran, Iran.
Abstract We derive a priori bounds for positive supersolutions of −∆p u = ρ(x)f (u), where p > 1 and ∆p is the p-Laplace operator, in a smooth bounded domain of RN with zero Dirichlet boundary conditions. We apply the results to nonlinear elliptic eigenvalue problem −∆p u = λf (u), with Dirichlet boundary condition, where 1 f is a nondecreasing continuous differentiable function on [0, ∞] such that f (0) > 0, f (t) p−1 is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter λ∗p . In particular, we consider the nonlinearities f (u) = eu and f (u) = (1 + u)m (m > p − 1 ) and give explicit estimates on λ∗p . As a by-product of our results, we obtain a lower bound for the principal eigenvalue of the p-Laplacian that improves obtained results in the recent literature for some range of p and N . Key words: Nonlinear eigenvalue problem, Estimates of principal eigenvalue, Extremal parameter.
1. Introduction Let Ω be a smooth bounded domain of RN and p > 1. We consider the nonlinear elliptic problem −∆p u = ρ(x)f (u) x ∈ Ω, u>0 x ∈ Ω, (1.1) u=0 x ∈ ∂Ω
where ∆p is the p-Laplace operator defined by ∆p u := div(|∇u|p−2 ∇u), ρ : Ω → R is a nonnegative bounded function that is not identically zero and f satisfies (C) f : Df = [0, af ) → R+ := [0, ∞) (0 < af 6 +∞) is a nondecreasing C 1 function with f (u) > 0 for u > 0. We say that u is a solution of (1.1) if u ∈ W01,p (Ω), u ≥ 0 a.e., ρ(x)f (u) ∈ L1 (Ω), and Z Z p−2 ρ(x)f (u)ϕ, for all ϕ ∈ Cc∞ (Ω), |∇u| ∇u.∇ϕ = Ω
Ω
∞
that is, for all C functions ϕ with compact support in Ω. Note that, since u is p-superharmonic we have that if u 6≡ 0 then u > 0 a.e. in Ω, by the strong maximum principle (see [9, 23, 25, 26]). A solution u ∈ W01,p (Ω) is called a regular solution of (1.1) if ρ(x)f (u) ∈ L∞ (Ω). By the well known regularity results ¯ for some α ∈ (0, 1] (see for degenerate elliptic equations, if u is a regular solution of (1.1) then u ∈ C 1,α (Ω) for instance [9, 22]). Also, we say that u ∈ W01,p (Ω) is a supersolution of (2.1) if ρ(x)f (u) ∈ L1 (Ω) and −∆p u ≥ ρ(x)f (u) in the weak sense. Reversing the inequality one defines the notion of subsolution. ∗ Corresponding
author: A. Aghajani. Tel. +9821-73913426. Fax +9821-77240472. Email addresses:
[email protected] (A. Aghajani a,b ),
[email protected] (A.M. Tehrani a ) September 20, 2016
The ball of radius R centered at x0 in RN will be denoted by BR (x0 ). Given a set Ω in RN we let |Ω| denote its N -dimensional Lebesgue measure. The p-torsion function ψ of a domain Ω is the unique solution of the problem −∆p u = 1 x ∈ Ω, u = 0 x ∈ ∂Ω. We shall denote ψM := supx∈Ω ψ(x). In this paper, first we consider C 1 positive supersolutions u of (1.1) in section 2 (by a positive solution we mean a solution which is nonnegative and nontrivial) and give explicit pointwise lower bounds for u under −1 the condition that f satisfies (C) and f p−1 ∈ L1 (0, a) for a ∈ (0, af ). In particular, we prove that F (u(x)) > where F (t) =
Z
0
t
ds f (s)
1 p−1
1 p − 1 ρx (dΩ (x))dΩ (x)p p−1 p N
, 0 < t < af ,
for all x ∈ Ω,
ρx (r) = inf ρ(y) : |y − x| < r , and dΩ (x) := dist(x, ∂Ω).
As an application, in section 3, we consider the eigenvalue problem −∆p u = λf (u) x ∈ Ω, u=0 x ∈ ∂Ω,
(1.2)
with f satisfies (C) and define the extremal parameter λ∗p as n o λ∗p = λ∗p (f, Ω) := sup λ > 0 : problem (1.2) has at least one positive bounded solution. . In the case when f , in addition, satisfies
1
(H) f : R+ → R+ is C 1 , f (0) > 0 and f (t) p−1 is superlinear at infinity (i.e., limt→∞
g(t) tp−1
= ∞),
X. Cabr´e and M. Sanch´ on in [[9], Theorem 1.4] proved that λ∗p ∈ (0, ∞) and for every λ ∈ (0, λ∗p ) problem (1.2) admits a minimal regular solution uλ . Minimal means that it is smaller than any other supersolution R ∞ ds 1 of the problem. If in addition f (t) p−1 is convex function satisfying 0 < ∞, then (1.2) admits no 1 f (s) p−1
solution for λ > λ∗p (f, Ω). Moreover, the family {uλ } is increasing in λ and every uλ is semi-stable in the sense that the second variation of the energy functional associated with (1.2) is nonnegative definite (see Definition 1.1 in [9]). Using this property in [9] the authors established that u∗ = limλ↑λ∗p uλ is a solution of (1.2) with λ = λ∗p whenever lim inf t→∞
tf ′ (t) f (t)
> p − 1; u∗ is called the extremal solution.
Let λ1 = λ(p, Ω) be the first eigenvalue of p-Laplacian subjected to Dirichlet boundary condition, i.e., R |∇v|p dx RΩ . (1.3) λ1 := min p 06=v∈W01,p (Ω) Ω |v| dx
Azorero and Peral in [1] showed that if f (u) = eu then λ∗p 6 max λ1 , λ1 [9] extended this result for every nonlinearity f satisfying (H), as n tp−1 o λ∗p 6 max λ1 , λ1 sup . t>0 f (t)
p−1 p−1 . p
Cabr´e and Sanch´ on in
(1.4)
In both proofs the authors (by a contradiction argument) used comparison principle for the p-Laplacian operator to construct, for every ε > 0 sufficiently small, an increasing sequence of functions whose limit is 2
in W01,p (Ω) and solves the problem −∆p w = (λ1 + ε)wp−1 , then used the fact that the first eigenvalue for the p-Laplacian is isolated to get a contradiction. Before presenting our estimates on λ∗p , first we improve (1.4) as the following (using the homogeneity property of p-Laplacian and (1.4) itself) tp−1 . (1.5) λ∗p 6 λ1 sup t>0 f (t) Then we prove the following upper bound, without using the fact that the first eigenvalue for the p-Laplacian is isolated, Z 1 ∞ ds p−1 ∗ λp 6 p−1 , 1 ψM 0 f (s) p−1 where ψM as defined before is the supremum (maximum) of the p-torsion function on Ω. As we shall see, in many cases, this represents a sharper upper bound than (1.5). While there is no explicit formula for the lower bound in the literature for the critical parameter λ∗p (p 6= 2), which is very important in application, we shall prove the following lower bound for the extremal parameter of problem (1.2 ) with general nonlinearity f satisfying C, using the method of sub-super solution, λ∗p > max
n 1 tp−1 p−1 sup f (t) , ψM 0 , ρx dΩ (x) 1 p N p−1 F (u(x)) >
1 p − 1 ρx (dΩ (x))dΩ (x)p p−1 p N
|y − x| < dΩ (x).
for all x ∈ Ω.
Proof. First note that by the assumptions on f and definition of F we have F ′ (t) = F ′′ (t) =
(2.3)
−f ′ (t)
(2.4) 1 1
f (t) p−1
> 0 and
6 0, 0 < t < af , thus using Lemma 2.1 (with G = F and a = af ) and the fact that p (p − 1)f (t) p−1 u is a supersolution, we can write −∆p F (u) ≥ F ′ (u)p−1 (−∆p u) 1 = (−∆p u) f (u) ≥ ρ(x) = −∆p ψρ . Now since we have F (u) = ψρ = 0 on ∂Ω, then by the maximum principle we get F u(x) > ψρ (x) for every x ∈ Ω that proves (2.2). To prove (2.3) we need to estimate ψρ from below. Let x ∈ Ω. Then for y ∈ BdΩ (x) (x) we get from (2.1) (2.5) − ∆p ψρ (y) = ρ(y) > ρx dΩ (x) . Now consider the auxiliary function w(y) =
p−1 p
p p dΩ (x) p−1 − x − y p−1 1
N p−1
which satisfies −∆p w = 1 in
BdΩ (x) (x) and w = 0 on ∂BdΩ (x) (x). Then from (2.5) we get 1 −∆p ψρ (y) > −∆p ρx dΩ (x) p−1 w(y) ,
1 hence by the maximum principle ψρ (y) > ρx dΩ (x) p−1 w(y) in BdΩ (x) (x) that with the aid of (2.2) proves (2.3). Taking y = x in (2.3) gives (2.4). 3. Application to eigenvalue problem 3.1. Lower and upper bounds for λ∗p (f, Ω) Consider the nonlinear eigenvalue problem (1.2). Before presenting our results based on Theorem 2.1, first we improve the upper bound (1.4) for the extremal parameter λ∗p (f, Ω) with f satisfies (H), in the following lemma using the homogeneity property of p- Laplacian and (1.4) itself. Lemma 3.1. For the extremal parameter of problem (1.2) with f satisfies (H), we have λ∗p 6 λ1 sup t>0
5
tp−1 . f (t)
(3.1)
Proof. Assume that for some λ > 0, uλ be the minimal solution of (1.2) and take an arbitrary positive number M ∈ (0, ∞). Then it is easy to see that the function w := M uλ is a bounded solution of the equation −∆p w = M p−1 λg(w) x ∈ Ω, w=0 x ∈ ∂Ω, u where g(u) := f ( M ). Hence from (1.4) we must have
n tp−1 o . M p−1 λ 6 max λ1 , λ1 sup t>0 g(t) However, we have sup t>0
(3.2)
tp−1 tp−1 = M p−1 sup , thus from (3.2) we get g(t) t>0 f (t) λ 6 max
n λ tp−1 o 1 . , λ1 sup p−1 M t>0 f (t)
(3.3)
Now for M sufficiently large we get from (3.3) that λ 6 λ1 sup t>0
tp−1 , f (t)
which proves (3.1). Theorem 3.1. Let λ∗p be the extremal parameter of problem (1.2) with f satisfy (C). Then λ∗p and λ∗p > max
6
Z 1
p−1 ψM
af
0
n 1 tp−1 , sup p−1 0
p p − 1 |Ω| p−1 ( ) . 2p − 1 P (Ω)
Hence from Theorem 3.1 we get the following explicit bounds for λ∗p . 9
(3.13)
Corollary 3.1. Let λ∗p be the extremal parameter of problem (1.2) with f satisfy C. Then Z af p p−1 2p N tp−1 ds p−1 ∗ , sup 6 λ 6 θ p,Ω 1 p p p−1 diam(Ω) 0
N p , pR
p ∈ (1, ∞).
(3.15)
The lower bound (3.15) becomes sharp when p → 1, as it is shown by V. Friedman and B. Kawhol in [15] that λ1 (Ω) converges to the Cheeger constant h(Ω) when p → 1. However, it is not sharp when p → ∞, as from [20] we know that 1 1 lim (λ1 ) p (Ω) = , p→∞ rΩ 1
where rΩ is defined in (3.11). Hence, limp→∞ (λ1 ) p (Ω) = R1 , while the p-th root of the right hand side of (3.15) goes to zero when p → ∞. Here, we give some lower bounds for λ1 using our results. First note that from (3.8) and (3.12) we have p 2 1 p p−1 N. (3.16) λ1 (Ω) > p−1 > p−1 diam(Ω) ψM In particular, in the special case when Ω is the ball BR then λ1 (BR ) >
1 p−1 ψM
>
p p−1 N , p−1 Rp
(3.17)
which is recently obtained by J. Benedikt and P. Der´abek in [3].
2p−1 p−1
The lower bound (3.17) is better than (3.15) when N < pp−1 , and also becomes sharp in both critical cases p → 1 and p → ∞. Also, the following lower bound for λ1 , which is a consequence of Example 2.1 and (3.1), gives better bound on λ1 (B), for more values of p and N . 2p−1 p p−1 ( p )p−1 N N 6 p−1 e(p−1) , 2p−1 (3.18) p p−1 p2 λ1 (B) > e p−1 N p ( ) p p e(p−1) < N 6 p−1 , ( pe )p−1 (N − p) N > p2 . p−1
p−1
10
Benedikt and Der´abek in [4] also presented upper and lower bounds for λ1 (Ω) on a bounded domain Ω ⊂ RN . In particular, when Ω = B they proved that λ1 (B) > N p.
(3.19)
Comparing (3.18) and (3.19), one can easily check that when 1 < p 6 2 the lower bound (3.18) is better p+1 p p−1 than (3.19) in every dimension N . Also, when p > 2 the same is true when N > . e 4. Nonexistence results Here we show that how one can apply Theorem 2.1 to prove nonexistence of positive solutions of differential inequalities involving p-Laplacian. Consider the differential inequality −∆p u > λρ(x)f (u) x ∈ Ω, u>0 x ∈ Ω, (4.1) u ∈ W01,p (Ω). Theorem 4.1. Let f satisfy (C), and ρ : Ω → R is a nonnegative function that is not identically zero. Then i) Inequality (4.1) has no positive C 1 solution if λ>
N ||F ||p−1 p p−1 ∞ . p−1 supx∈Ω ρx dΩ (x) dΩ (x)p
(4.2)
ii) If ρ(x) = |x|α , α > 0 and Ω = BR , then the same is true if λ>
α + p p−1
||F ||∞
p−1
(α + N )R−(α+p) .
Proof. i) If (4.1) has a positive solution u, then from (2.4) in Theorem 2.1 (by replacing f with λf ) we get Z
0
u(x)
ds f (s)
1 p−1
p−1
>λ
p − 1 p−1 ρx dΩ (x) dΩ (x)p , p
x ∈ Ω,
and taking supremum on both sides over Ω we arrive at a contradiction with (4.2). ii) Now, let ρ(x) = |x|α and Ω = BR . In this case we can use (2.2) directly. Indeed, it is easy to see that the function α+p α+p ψρ (x) = C R p−1 − |x| p−1 ,
with
C :=
−1 p−1 α + N p−1 , α+p
is the solution of (2.1) with ρ(x) = |x|α , hence from (2.2) we must have 1
F (u(x)) > λ p−1 ψρ (x), x ∈ BR . Taking supremum over BR we get the desired result. As an application of this result, consider the eigenvalue problem ( |x|α −∆u = λ (1−u) x ∈ Ω, 2 u=0 x ∈ ∂Ω, 11
that in dimension N = 2 models a simple Micro-Electromechanical-Systems MEMS device, see [11, 13, 18, 19]. Let λ∗ (called pull-in voltage) be the extremal parameter of the above eigenvalue problem, then from Theorem 2.7, we have λ∗ 6
(α + 2)(α + N ) −(α+2) R . 3
This upper bound substantially improve the ones obtained in [2, 18, 19]. It could be interesting to compare this bound to the lower bound for λ∗ given in [13], then we have max
n 4(α + 2)(α + N ) (α + 2)(3N + α − 4) o (α + 2)(α + N ) −(α+2) R−(α+2) 6 λ∗ 6 , R . 27 9 3
5. Acknowledgement This research was in part supported by a grant from IPM (No. 94340123). References [1] B. Abdellaoui, I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential, Ann. Mat. Pura Appl., (4) 182 (2003), 247-270. [2] A. Aghajani, A. Mosleh Tehrani, N. Ghoussoub, Pointwise lower bounds for solutions of semilinear elliptic equations and applications, Adv. Nonlinear Stud., 14 (2014), 839-856. [3] J. Benedikt, P. Dr´ abek, Asymptotics for the principal eigenvalue of the p-Laplacian on the ball as p approaches 1, Nonlinear Anal. 93 (2013) 23-29. [4] J. Benedikt, P. Dr´ abek, Estimates of the principal eigenvalue of the p-Laplacian, J. Math. Anal. Appl. 393 (2012) 311-315. [5] J. Benedikt, P. Dr´ abek, P. Girg, The second eigenfunction of the p-Laplacian on the disk is not radial, Nonlinear Anal. 75 (2012), 4422-4435. [6] H. Brezis, X. Cabr´ e, Some simple nonlinear PDE’s without solutions, Bull UMI, 1 (1998), 223-262. [7] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow-up for ut −∆uu = g(u) revisited, Ad. Diff. Eq. 1 (1996), 73-90. [8] X. Cabr´ e, A. Capella, M. Sanch´ on, Regularity of radial minimizers of reaction equations involving the p-Laplacian, Calc. Var. Partial Differential Equations 34 (2009), 475-494. [9] X. Cabr´ e, M. Sanch´ on, Semi-stable and extremal solutions of reaction equations involving the p-Laplacian, Comm. Pure Appl. Anal., 6 (2007), 43-67. [10] J. Cheeger, A lower bound for the smalest eigenvalue of the Laplacian, in: Problems in Analysis, A Symposium in Honor of Salomon Bochner, Ed.: R.C. Gunning, Princeton Univ. Press (1970), pp. 195-199. [11] C. Cowan, N. Ghoussoub, Estimates on pull-in distances in microelectromechanical systems models and other nonlinear eigenvalue problems, SIAM J. Math. Anal., 42 (2010), 1949-1966. [12] F. Della Pietra, N. Gavitone. Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators, Math. Nachr., 2013. [13] P. Esposito, N. Ghoussoub, Y.J. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, volume 20 of Courant Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences, New York, 2010. [14] A. Ferrero, On the solutions of quasilinear elliptic equations with a polynomial-type reaction term, Adv. Differential Equations 9 (2004), 1201-1234. [15] V. Fridman, B. Kawohl, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carol., 44 (2003), 659-667. [16] J. Garc´ıa-Azorero, I. Peral Alonso, On an Emden-Fowler type equation, Nonlinear Anal., 18 (1992), 1085-1097. [17] J. Garc´ıa-Azorero, I. Peral Alonso, J.P. Puel, Quasilinear problems with exponential growth in the reaction term, Nonlinear Anal., 22 (1994), 481-498. [18] N. Ghoussoub, Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2006), 1423-1449. [19] Y. Guo, Z. Pan, M.J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math., 66 (2005), no. 1, 309–338. [20] P. Juutinen, P. Lindqvist, J.J. Manfredi, The 1-eigenvalue problem, Arch. Rational Mech. Anal., 148 (1999), 89-105. [21] L. Lefton, D. Wei , Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method, Numer. Funct. Anal. Optim., 18 (1997) 389-399. [22] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic e quations, Nonlinear Anal. 11 (1988), 1203-1219. [23] M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations , Nonlinear Anal. 37 (1999), 431-448. [24] I. Peral, Multiplicity of solutions for the p-Laplacian , International Center for Theoretical Physics Lecture Notes, Trieste, 1997, http://www.uam.es/personal-pdi/ciencias/ireneo.
12
[25] N.S. Trudinger, On Harnack type inequalities and their application to quasi linear elliptic equations , Comm. Pure Appl. Math. 20 (1967), 721-747. [26] J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202.
13
