2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 9–22. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Semi-classical analysis and vanishing properties of solutions to quasilinear equations ^{∗}

Yves Belaud

Abstract

Let Ω be an open bounded subset of ℝ^{N} and b a measurable nonnegative function in Ω. We deal with the time compact support property for

u_{t} − Δu + b(x)|u|^{q−1}u = 0

for p ≥ 2 and

u_{t} − div(|∇u|^{p−2}∇u) + b(x)|u|^{q−1}u = 0

with m ≥ 1 where 0 ≤ q < 1. We give criteria associated to the first eigenvalue of some quasilinear Schrödinger operators in semi-classical limits. We also provide a lower bound for this eigenvalue.

1 Introduction

Let Ω be a regular bounded domain of ℝ^{N} (N ≥ 1) and q ∈ [0, 1). We consider the weak solution of the degenerate parabolic equations subject to the Neumann boundary condition:

u_{t} − Δu + b(x)|u|^{q−1}u = 0 in Ω × (0, ∞),
∂_{ν}u = 0 on ∂Ω,
u(x, 0) = u_{0}(x) in Ω,

(1.1)

and more generally,

u_{t} − div(|∇u|^{p−2}∇u) + b(x)|u|^{q−1}u = 0 in Ω × (0, ∞),
∂_{ν}u = 0 on ∂Ω,
u(x, 0) = u_{0}(x) in Ω,

(1.2)

∗Mathematics Subject Classifications: 35K55, 35P15.

Key words: evolution equations, p-Laplacian, porous-medium, strong absorption, regularizing effects, semi-classical limits. \begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University. Published October 21, 2002.

with p ≥ 2, or

u_{t} − Δ(u^{m}) + b(x)|u|^{q−1}u = 0 in Ω × (0, ∞),
∂_{ν}u = 0 on ∂Ω,
u(x, 0) = u_{0}(x) in Ω,

(1.3)

with m ≥ 1.

Many authors have already dealt with such equations giving a wide range of applications in physical mathematics. Now, our task is to describe a compact compact support property, in time.

Definition. A solution u satisfies the Time Compact Support property (for short TCS property) if there exists a time T such that for all t ≥ T and all x ∈ Ω, u(x, t) = 0.

First, we study some simple cases for (1.1):

1) Suppose that there exists a real γ such as b(x) ≥ γ > 0 a.e. in Ω. From the maximum principle, u(x, t) ≤ (1 − γ(1 − q)t)^{\frac{1}{1−q}} in Ω × (0, ∞). The nonlinear absorption is stronger than the diffusion and the TCS property holds.

2) We have a different feature if we assume that there exists a connected open set ω such as b(x) = 0 a.e. in ω (no absorption in ω). Then usually, u has not the compact support property. Indeed, if we denote by λ(ω) the first eigenvalue of −Δ in W_{0}^{1,2}(ω) and ζ the first eigenfunction with ∥ζ∥_{L^{∞}(ω)} = 1 and ζ ≥ 0, then from the maximum principle, u(x, t) ≥ ζ(x) e^{−λ(ω)t} for all x in ω and for all t ≥ 0.

Up to some minor changes, the previous examples are also valid if u satisfies (1.2) and (1.3). The compact support property is related to {x : b(x) = 0} and the behaviour of the function b in a neighbourhood of this set.

2 The time compact support property

The starting idea was in the article of Kondratiev and Véron [7]. They established this property for (1.1) with the help of the following quantities

μ_{n} = inf { ∫_{Ω}(|∇v|^{2} + 2^{n}b(x)|v|^{2})dx : v ∈ W^{1,2}(Ω), ∫_{Ω} |v|^{2} dx = 1},

with n positive integer number. More precisely, up to a small change, they proved the following theorem.

Theorem 2.1 Suppose that u is a solution of (1.1) and

∑_{n=0}^{+∞} \frac{ln μ_{n}}{μ_{n}} < +∞,

then there exists some T > 0 such that u(x, t) = 0 for (x, t) ∈ Ω × [T, +∞).

We see that μ_{n} are linked to well-known questions in the semi-classical limit of Schrödinger operator of the type −Δ + 2^{n}b(.).

In [3], the authors give a first extension of this theorem by replacing the sequence 2^{n} by any decreasing sequence going to zero. For the sake of simplicity, we denote by μ(α) the lowest eigenvalue of the Neumann realization of the Schrödinger operator −Δ + α^{q−1}b(.) in W^{1,2}(Ω), that is,

μ(α) = inf { ∫_{Ω}(|∇v|^{2} + α^{q−1}b(x)|v|^{2})dx : v ∈ W^{1,2}(Ω), ∫_{Ω} |v|^{2} dx = 1}. (2.1)

They proved the following theorem [3, page 50].

Theorem 2.2 Assume that (α_{n}) is a decreasing sequence of positive numbers such that

∑_{n=1}^{+∞} \frac{1}{μ(α_{n})}( ln(μ(α_{n})) + ln(\frac{α_{n}}{α_{n+1}} ) + 1) < +∞,  (2.2)

then any solution of (1.1) satisfies the TCS property.

The proof is based on an iterative method using the following lemma.

Lemma 2.1 Suppose that b ≥ 0 a.e. in Ω, 0 ≤ q < 1 and u is a bounded weak solution of (1.1) such that ∥u_{0}∥_{L^{∞}(Ω)} ≤ α for some α > 0. Then

∥u(., t)∥_{L^{∞}(Ω)} ≤ min (1, C(μ(α))^{N/4}e^{−tμ(α)})∥u_{0}∥_{L^{∞}(Ω)},  (2.3)

where C = C(Ω) is a positive real number.

Outline of the proof. We use u as test-function and since u^{1−q} ≥ α^{1−q}, we have

\frac{1}{2} \frac{d}{dt} ∫_{Ω} u^{2} dx + ∫_{Ω}(|∇u|^{2} + bα^{q−1}u^{2}) dx ≤ 0.

The definition of μ(α) and Hölder’s inequality gives

∥u(., s)∥_{L^{2}(Ω)} ≤ e^{−sμ(α)}|Ω|^{1/2}∥u_{0}∥_{L^{∞}(Ω)},

for all positive real number s. The regularizing effects associated to this type of equation can be write under the following form [11, 12]:

∥u(., t)∥_{L^{∞}(Ω)} ≤ C(1 + \frac{1}{t − s})^{N/4}∥u(., s)∥_{L^{2}(Ω)},

for all t > s. Taking t − s = 1/μ(α) completes the proof of the lemma. □

Sketch of the proof of the theorem 2.2. (α_{n}) is a decreasing sequence which tends to zero. We shall construct an increasing sequence (t_{n}) such that for all n,

∀t ≥ t_{n}, ∥u(., t)∥_{L^{∞}(Ω)} ≤ α_{n}.

If lim_{n→+∞} t_{n} = T < +∞ then u satisfies the TCS property. To do this, we use an iterative method to find an upper bound for ∑_{n} t_{n+1} − t_{n} under the form of a convergent series. We set t_{0} = 0 and α = α_{0} = ∥u_{0}∥_{L^{∞}(Ω)}. Applying Lemma 2.1 gives an upper bound for ∥u(., t)∥_{L^{∞}(Ω)}. t_{1} is defined by

C(μ(α_{0}))^{N/4}e^{−(t_{1}−t_{0})μ(α_{0})}α_{0} = α_{1}.

A this point, we apply Lemma 2.1 but for time t ≥ t_{1} with α = α_{1}. Iterating this process provide us the formula

C(μ(α_{n}))^{N/4}e^{−(t_{n+1}−t_{n})μ(α_{n})}α_{n} = α_{n+1}.

So we obtain an upper bound for the series ∑_{n} t_{n+1} − t_{n}. □

An analoguous result can be proved for (1.2). But before, we recall the regularizing effects for this type of equation [11, 12].

Theorem 2.3 Let p > 1. Suppose that u is a weak solution of

u_{t} − div(|∇u|^{p−2}∇u) + B(x, t, u) = 0 in Ω × (0, ∞),
∂_{ν}u = 0 on ∂Ω,
u(x, 0) = u_{0}(x) ∈ L^{r}(Ω),

where B is a Caratheodory functions which satisfies B(x, t, ρ)ρ ≥ 0 a.e. in Ω × (0, ∞). If r ≥ 1, r > N(2/p − 1) then

∥u(., t)∥_{L^{∞}(Ω)} ≤ C(1 + \frac{1}{t})^{δ(r)}∥u(., 0)∥_{L^{r}(Ω)}^{σ(r)},

with C = C(Ω, p), δ(r) = \frac{N}{rp+N(p−2)} and σ(r) = \frac{rp}{rp+N(p−2)}.

In a similar way, we introduce

μ(α, p) = inf { ∫_{Ω}(|∇v|^{p} + α^{q−(p−1)} b(x) |v|^{p})dx : v ∈ W^{1,p}(Ω), ∫_{Ω} |v|^{p} dx = 1}.

Here μ(α, p) is the first eigenvalue in W^{1,p}(Ω) for the Neumann boundary condition of

u ↦ −Δ_{p}u + α^{q−(p−1)}b(.)u^{p−1}.

The theorem states as follows [1]:

Theorem 2.4 Let 0 ≤ q < 1, p > 2 and assume that there exist two sequences of positive real numbers (α_{n}) and (r_{n}) such that (α_{n}) is decreasing and

∑_{n=0}^{∞} \frac{r_{n}^{p−1}}{α_{n+1}^{p−2}μ(α_{n}, p)^{σ(r_{n})}} < +∞.  (2.4)

Then any solution of (1.2) with initial bounded data satisfies the TCS property.

Consequently, if r_{n} = ln μ(α_{n}, p), we have the following statement.

Corollary 2.1 Under the same assumptions on q and p, if there exists a decreasing sequence of positive real numbers (α_{n}) such that

∑_{n=0}^{∞} \frac{(ln μ(α_{n}, p))^{p−1}}{α_{n+1}^{p−2}μ(α_{n}, p)} < +∞,  (2.5)

then any solution of (1.2) satisfies the TCS property.

Theorem 2.4 comes from the following lemma.

Lemma 2.2 Suppose there exists a measurable function u in Ω × ℝ^{+} which satisfies weakly (1.2) with ∥u_{0}∥_{L^{∞}(Ω)} ≤ α for some α > 0. Then

∥u(., t)∥_{L^{r}(Ω)} ≤ (\frac{1}{∥u(., 0)∥_{L^{r}(Ω)}^{2−p} + C_{1}μ(α, p)t})^{\frac{1}{p−2}} ,  (2.6)

where C_{1} = C_{1}(Ω, r, p) is a positive real constant and there exist two positive real numbers C = C(Ω, p) and C_{2} = C_{2}(r, p) such that

∥u(., t)∥_{L^{∞}(Ω)} ≤ min (C(1 + \frac{2}{t})^{δ(r)}(\frac{1}{∥u(., 0)∥_{L^{∞}(Ω)}^{2−p} + C_{2}μ(α, p)t})^{\frac{σ(r)}{p−2}} , 1),

with δ(r) = \frac{N}{rp+N(p−2)} and σ(r) = \frac{rp}{rp+N(p−2)}.

Idea in the proofs. The principle to prove them remains true. It is a bit more complicated because the term u_{t} is not homogenuous with u^{p−1} but it follows exactly the Kondratiev-Vron method as shown in the proof of Theorem 2.2. The main differences are technical. Instead of using u as test-function, we use u|u|^{r_{n}−1} at each step of the iteration. An estimate of the asymptotic behaviour when r → +∞ for the constant C_{2} = C_{2}(r, p) is needed. The proof of the theorem ends with sharp upper bounds for the series ∑_{n} t_{n+1} − t_{n}. □

Now, let us talk about equation 1.3. Formally, replacing p − 1 by m give the same results [11, 12]:

Theorem 2.5 Let m > 0 and u be a weak solution of

u_{t} − Δ(u^{m}) + B(x, t, u) = 0 in Ω × (0, ∞),
∂_{ν}u = 0 on ∂Ω,
u(x, 0) = u_{0}(x) ∈ L^{r}(Ω),

where B is a Caratheodory function satisfying B(x, t, ρ)ρ ≥ 0 a.e. in Ω×(0, ∞). If r ≥ 1 and r > N(1 − m)/2, then

∥u(., t)∥_{L^{∞}(Ω)} ≤ C(1 + \frac{1}{t})^{δ(r)}∥u(., 0)∥_{L^{r}(Ω)}^{σ(r)} ,

with C = C(Ω, m), δ(r) = \frac{N}{2r+N(m−1)} and σ(r) = \frac{2r}{2r+N(m−1)}.

We set quantities adapted to the problem μ′(α, m) = inf { ∫_{Ω}(|∇v|^{2} + α^{q−m}b(x)|v|^{2})dx : v ∈ W^{1,2}(Ω), ∫_{Ω} |v|^{2} dx = 1}. Thus, Theorem 2.6 ([1]) Let 0 ≤ q < 1, m > 1 and assume that there exists two sequences of positive real numbers (α_{n}) and (r_{n}) such that (α_{n}) is decreasing and

∑_{n=0}^{∞} \frac{r_{n}^{m}}{α_{n+1}^{m−1}μ′(α_{n}, m)^{σ(r_{n})}} < +∞.  (2.7)

Then any solution of (1.3) with initial bounded data satisfies the TCS property.

With r_{n} = ln μ′(α_{n}, m), we deduce the following statement. Corollary 2.2 Under the above assumptions on q and m, if there exists a decreasing sequence of positive real numbers (α_{n}) such that

∑_{n=0}^{∞} \frac{(ln μ′(α_{n}, m))^{m}}{α_{n+1}^{m−1}μ′(α_{n}, m)} < +∞,

then any solution of (1.3) satisfies the TCS property.

The proof of Theorem 2.6 also comes from the following lemma. Lemma 2.3 We suppose there exists a measurable function u in Ω × ℝ^{+} which satisfies weakly (1.3) with ∥u_{0}∥_{L^{∞}(Ω)} ≤ α for some α > 0. Then

∥u(., t)∥_{L^{r}(Ω)} ≤ (\frac{1}{∥u(., 0)∥_{L^{r}(Ω)}^{1−m} + C_{1}μ′(α, m) t})^{1/(m−1)},  (2.8)

with C_{1} = C_{1}(Ω, r, m) and there exist two positive real numbers C = C(Ω, m) and C_{2} = C_{2}(r, m) such that

∥u(., t)∥_{L^{∞}(Ω)} ≤ min (C(1 + \frac{2}{t})^{δ(r)}(\frac{1}{∥u(., 0)∥_{L^{∞}(Ω)}^{1−m} + C_{2}μ′(α, m)t})^{\frac{σ(r)}{m−1}} , 1),

where δ(r) and σ(r) are defined in Theorem 2.5

The assumptions in Theorem 2.2 and Corollaries 2.1, 2.2 admit a simpler form. A comparaison between series and integral gives the following theorem. Theorem 2.7 (Integral criterion [3, 1]) Let 0 ≤ q < 1. 1) If p ≥ 2 and

∫_{0}^{1} \frac{(ln μ(t, p))^{p−1}}{t^{p−1}μ(t, p)} dt < +∞,

then all solutions of (1.2) satisfy the TCS property.

2) If m ≥ 1 and

∫_{0}^{1} \frac{(ln μ′(t, m))^{m}}{t^{m}μ′(t, m)} dt < +∞,

then all solutions of (1.3) satisfy the TCS property.

We remark that μ(t) = μ(t, 2) and that (1.1) is a particular case of (1.2) for p = 2 and (1.3) for m = 1. The proof is first establish for p = 2 [3, page 51] and then for p > 2 and m > 1 [1]. What is remarkable is that this criterion has a same simple form in all cases.

For applications, μ(t, p) and μ′(t, m) have to be linked directly to the function b. We recall that μ(α, p) is the first eigenvalue in W^{1,p}(Ω) for the Neumann boundary condition of u ↦ −Δ_{p}u + α^{q−(p−1)}b(.)u^{p−1}.

The aim of semi-classical analysis is to describe the behavior of the spectrum of the operator u ↦ −Δ_{p}u + h^{−p}V (.)u^{p−1} in particular λ_{1}(h) the lowest eigenvalue. V is a function which holds in our case

V ∈ L^{∞}(Ω), ess inf_{Ω} V = 0 and ∫_{Ω} V (x) dx > 0.  (2.9)

We denote by γ a positive number which satisfies:

γ \begin{cases}= \frac{N}{p} & for 1 < p < N, \\ ∈ (1, +∞) & for p = N, \\ = 1 & for p > N, \end{cases}  (2.10)

Corollary 2.3 If (2.9) holds then for h small enough,

λ_{1}(h)(meas{x : V (x) ≤ h^{p}λ_{1}(h)})^{1/γ} ≥ C,  (2.11)

where C = C(p, N, γ, Ω, V) is a positive constant.

μ(t, p) can be written as μ(t, p) = λ_{1}(t^{\frac{(p−1)−q}{p}}) which after a change of variables gives

∫_{0}^{1} \frac{(ln μ(t, p))^{p−1}}{t^{p−1}μ(t, p)} dt = ∫_{0}^{1} \frac{(ln λ_{1}(h))^{p−1}}{h^{\frac{p(p−1)−(1+q)}{p−(1+q)}} λ_{1}(h)}dh.

If we have an estimate of the type

λ_{1}(h) ≥ C \frac{1}{h^{θ}} ,

where C and θ are two positive real numbers, then the integral criterion holds for p > 2 provided

θ > \frac{p(p − 2)}{p − (1 + q)}.  (2.12)

Similar expressions can be found for p = 2 and m > 1. Finally, we obtain next theorem.

Theorem 2.8 (1/b criterion [3, 1]) Let 0 ≤ q < 1 and b be a bounded measurable function such that

ess inf_{Ω} b = 0 and ∫_{Ω} b(x) dx > 0.

1) If p = 2 and ln(1/b) ∈ L^{s}(Ω) for some s > N/2 then equation (1.1) satisfies the TCS property.

2) If p > 2 and (1/b)^{s} ∈ L^{1}(Ω) for some s with

s > \begin{cases}\frac{p−2}{1−q}(\frac{N}{p}) & for p ≤ N, \\ \frac{p−2}{1−q} & for p > N, \end{cases}

then equation (1.2) satisfies the TCS property.

3) If m > 1 and (1/b)^{s} ∈ L^{1}(Ω) for some s with

s > \begin{cases}\frac{m−1}{1−q}(\frac{N}{2} ) & for N ≥ 2, \\ \frac{m−1}{1−q} & for N = 1, \end{cases}

then equation (1.3) satisfies the TCS property.

Outline of the proof. the three cases are based on Marcinkiewicz type inequalities. For 1)

meas \{x ∈ Ω : ln \frac{1}{b(x)} ≥ ln \frac{1}{h^{2}λ_{1}(h)}\} ≤ \frac{1}{( ln \frac{1}{h^{2}λ_{1}(h)})^{s}} ∫_{Ω} ( ln \frac{1}{b(x)})^{s}dx,

and for 2)

meas \{x : \frac{1}{b(x)} ≥ \frac{1}{h^{p}λ_{1}(h)}\} ≤ (h^{p}λ_{1}(h))^{s} ∫_{Ω} (\frac{1}{b(x)})^{s}dx.

The proof ends with estimates such as (2.12) and some technical arguments. □

Remark 2.1 In the case where p = 2 and N ≤ 2, estimate (2.11) is not enough sharp so we use the formula of Lieb and Thirring. See [3] for details.

Now we apply the previous theorem to the radial functions.

Corollary 2.4 Suppose that 0 ∈ Ω. 1) If b(x) = exp(−\frac{1}{∥x∥^{β}}) with β < 2 then any solution of (1.1) satisfies the TCS property.

2) If b(x) = ∥x∥^{β} with p ≤ N and β < p(1 − q)/(p − 2) then any solution of (1.2) satisfies the TCS property. One has the same conclusion if p > N and β < N(1 − q)/(p − 2).

3) If b(x) = ∥x∥^{β} with N ≥ 2 and β < 2(1 − q)/(m − 1) then any solution of (1.3) satisfies the TCS property. One has the same conclusion if N = 1 and β < (1 − q)/(m − 1).

3 A lower bound for the first eigenvalue

This section is dedicated to estimating the first eigenvalue, in W^{1,p}(Ω), of the operator u ↦ −Δ_{p}u + h^{−p}V (.)u^{p−1}. We have seen that a lower bound is fundamental for applications. First,we introduce a sequence of definitions. We consider a non-empty connected open subset Ω ⊂ ℝ^{N} and a mesurable function V defined in Ω. We set

W^{1,p,V} (Ω) = {ψ ∈ W^{1,p}(Ω) : V (x)|ψ^{p}| ∈ L^{1}(Ω)}.

If W^{1,p,V} (Ω) ≠ {0} and ψ ∈ W^{1,p,V} (Ω), we set

F_{V} (ψ) = ∫_{Ω} |∇ψ|^{p} + V (x)|ψ|^{p} dx,  (3.1)

and define

λ_{1} = inf {F_{V} (ψ) : ψ ∈ W^{1,p,V} (Ω), ∫_{Ω} |ψ|^{p} dx = 1} ,  (3.2)

and for h > 0,

λ_{1}(h) = inf {F_{h^{−p}V} (ψ) : ψ ∈ W^{1,p,V} (Ω), ∫_{Ω} |ψ|^{p} dx = 1} ,  (3.3)

Thus λ_{1}(h) is the first eigenvalue of the operator

u ↦ −Δ_{p}u + h^{−p}V (.)|u|^{p−2}u.  (3.4)

in W^{1,p,V} (Ω) with Neumann boundary condition if the infimum is achieved by a regular enough element of W^{1,p,V} (Ω) and ∂Ω C^{1}. We start with a simple result which enlights our arguments. On the contrary to the linear case (p = 2), our proof is not based on the theory of pseudodifferential operators but on the continuous injections of W^{1,p}(Ω) into the L^{s} spaces for suitable s.

Theorem 3.1 Suppose N > p > 1. Then either λ_{1} = −∞ or

( ∫_{V (x)≤λ_{1}}(λ_{1} − V (x))^{N/p} dx)^{p/N} ≥ C(p, N),  (3.5)

where C = C(p, N) > 0 is the positive constant of the Sobolev inequality. In addition, if there exists a minimizer in W^{1,p,V} (ℝ^{N}),

( ∫_{V (x)<λ_{1}}(λ_{1} − V (x))^{N/p} dx)^{p/N} ≥ C(p, N).  (3.6)

Proof. Let ψ be in W^{1,p,V} (ℝ^{N}) with ∥ψ∥_{L^{p}(ℝ^{N})} = 1 then

∫_{ℝ^{N}} |∇ψ|^{p} dx + ∫_{ℝ^{N}} V (x)|ψ|^{p} dx = F_{V} (ψ) = F_{V} (ψ) ∫_{ℝ^{N}} |ψ|^{p} dx.

The integral with V is split in two parts, that is, ℝ^{N} = {x : V (x) < F_{V} (ψ)} ∪ {x : V (x) ≥ F_{V} (ψ)}. Therefore,

∫_{ℝ^{N}} |∇ψ|^{p} dx ≤ ∫_{V (x)<F_{V} (ψ)}(F_{V} (ψ) − V (x))|ψ|^{p} dx.  (3.7)

Hölder’s inequality leads to

∫_{ℝ^{N}} |∇ψ|^{p} dx

≤ ( ∫_{V (x)<F_{V} (ψ)}(F_{V} (ψ) − V (x))^{N/p} dx)^{p/N}( ∫_{ℝ^{N}} |ψ|^{p^{∗}} dx)^{1− \frac{p}{N}} . (3.8)

since {x : V (x) < F_{V} (ψ)} ⊂ ℝ^{N}. Non zero constants do not belong to W^{1,p,V} (ℝ^{N}) and so all functions ψ satisfy ∫_{ℝ^{N}} |∇ψ|^{p} dx > 0. We can apply Sobolev inequality. The Beppo-Levi theorem completes the proof. □

Remark 3.1 If Ω is any open domain of ℝ^{N}, we define

W_{0}^{1,p,V} (Ω) = {ψ ∈ W_{0}^{1,p}(Ω) : V (x)|ψ^{p}| ∈ L^{1}(Ω)},

and if W_{0}^{1,p,V} (Ω) ≠ {0},

λ̃_{1} = inf {F_{V} (ψ) : ψ ∈ W_{0}^{1,p,V} (Ω), ∫_{Ω} |ψ|^{p} dx = 1},

then the estimates in Theorem 3.1 hold for λ̃_{1}.

When Ω is a C^{1} bounded domain of ℝ^{N} and V is a measurable function such that

V ∈ L^{∞}(Ω), ess inf_{Ω} V = 0 and ∫_{Ω} V (x) dx > 0,  (3.9)

we set u_{h} the first eigenfunction related to the first eigenvalue λ_{1}(h). Recall that γ is a positive number which satisfies

γ \begin{cases}= \frac{N}{p} & for 1 < p < N, \\ ∈ (1, +∞) & for p = N, \\ = 1 & for p > N, \end{cases}  (3.10)

with \frac{γ}{γ−1} = +∞ if γ = 1. This γ is such that W^{1,p} imbeds L^{q}(Ω) continuously with q = p\frac{γ}{γ−1}.

Theorem 3.2 Assume that (3.9) holds. Then for h small enough,

( ∫_{V (x)<h^{p}λ_{1}(h)} (λ_{1}(h) − \frac{V (x)}{h^{p}})^{γ} dx)^{1/γ} ≥ C,

where C = C(p, N, γ, Ω, V) is a positive real constant.

Proof. We start with (3.8) because the beginning is similar. Replacing ℝ^{N}, ψ and V by Ω, u_{h} and \frac{V}{h^{p}} the Hölder’s inequality gives

∫_{Ω} |∇u_{h}|^{p} dx ≤ ( ∫_{V (x)<h^{p}λ_{1}(h)} (λ_{1}(h) − \frac{V (x)}{h^{p}})^{γ} dx)^{1/γ}( ∫_{Ω} |u_{h}|^{q} dx)^{p/q},

where q = p\frac{γ}{γ−1}. Thus, by the imbeddings,

( ∫_{V (x)<h^{p}λ_{1}(h)} (λ_{1}(h) − \frac{V (x)}{h^{p}})^{γ} dx)^{1/γ} ≥ C \frac{∥∇u_{h}∥_{L^{p}(Ω)}^{p}}{1 + ∥∇u_{h}∥_{L^{p}(Ω)}^{p}} ,

with C = C(p, N, Ω, γ) a positive real number. The main idea is to prove that

lim inf_{h→0} ∥∇u_{h}∥_{L^{p}(Ω)} > 0.

Suppose that there exists a sequence (h_{n}) of positive real numbers which goes to zero such that

lim_{n→+∞} ∥∇u_{h_{n}}∥_{L^{p}(Ω)} = 0.

Hence (u_{h_{n}}) is bounded in W^{1,p}(Ω), so there exists a function u_{0} in W^{1,p}(Ω) such that, up to a subsequence, u_{h_{n}} ⇀ u_{0} weakly in W^{1,p}(Ω). Obviously, ∥∇u_{0}∥_{L^{p}(Ω)} = 0. Therefore, u_{0} = C where C is a real. Thanks to the Rellich-Kondrachov theorem, up to a subsequence, u_{h_{n}} → C strongly in L^{p}(Ω) so C = (\frac{1}{meas(Ω)})^{\frac{1}{p}}. We deduce that lim_{n→+∞} h_{n}^{p}λ_{1}(h_{n}) = ^{\frac{∫_{Ω} V (x) dx}{meas(Ω)}}. But from lemma 3.2 in [3], lim_{h→0} h^{p}λ_{1}(h) = 0 which leads to a contradiction. □

A simpler form is provided in the following corollary.

Corollary 3.1 If (3.9) holds then for h small enough,

λ_{1}(h)(meas{x : V (x) < h^{p}λ_{1}(h)})^{γ} ≥ C,

where C = C(p, N, γ, Ω, V).

We end this section by quoting a theorem. For Ω a domain of ℝ^{N} bounded or not, regular or not and V a mesurable function defined on Ω such that W^{1,p,V} (Ω) ≠ {0}, we define a well for a mesurable function V [1].

Definition. We say that V has a well in U if U is a C^{1} bounded, connected, non-empty open set of Ω and if there exists ψ_{0} ∈ W^{1,p,V} (Ω) with ∥ψ_{0}∥_{L^{p}(Ω)} = 1

such that ∫_{Ω} V (x)|ψ_{0}|^{p} dx < a = essinf_{Ω\U} V with meas(Ω\U) > 0.

The term of well generalizes the definition in [8].

Theorem 3.3 ([3]) If V has a well in U, for h small enough,

( ∫_{V (x)≤h^{p}λ_{1}(h)}(λ_{1}(h) − h^{−p}V (x))^{γ} dx)^{1/γ} ≥ C,

where C is a positive constant which does not depend on h.

In addition, if there exists a minimizer in W^{1,p,V} (Ω),

( ∫_{V (x)<h^{p}λ_{1}(h)}(λ_{1}(h) − h^{−p}V (x))^{γ} dx)^{1/γ} ≥ C.

The proof is technical but some arguments have already been used for Theorem 3.2.

4 Summary and open questions

For the sake of completeness, we quote another theorem of.

Theorem 4.1 ([3]) Suppose that b is a continuous and nonnegative function defined in \overline{Ω} which satisfies for some x_{0} ∈ Ω

lim_{r→0} r^{2} ln(1/∥b∥_{L^{∞}(B_{r}(x_{0}))}) = ∞.

If u is a weak solution of (1.1) then u does not satisfies the TCS property.

Up to now, we have the following:

p = 2  p > 2  m > 1

Integral criterion ∫_{0}^{1} \frac{ln μ(t)}{tμ(t)}dt < ∞ ∫_{0}^{1} \frac{(ln μ(t,p))^{p−1}}{t^{p−1}μ(t,p)} dt < ∞ ∫_{0}^{1} \frac{(ln μ′(t,m))^{m}}{t^{m}μ′(t,m)} dt < ∞

1/b criterion with

{ln(1/b) ∈ L^{s} \atop s > \frac{N}{2}} 1/b ∈ L^{s} ; s > \frac{p−2}{1−q} \frac{N}{p} , N ≥ p ; s > \frac{p−2}{1−p}, N < p 1/b ∈ L^{s} ; s > \frac{m−1}{1−q} \frac{N}{2} , N ≥ 2 ; s > \frac{m−1}{1−q}, N = 1

Radial case for β ≥ 0 and

{exp(−1/∥x∥^{β}) \atop β < 2}

∥x∥^{β} ; \frac{p(1−q)}{p−2}, N ≥ p ; β < \frac{N(1−q)}{p−2}, N < p ∥x∥^{β} ; β < \frac{2(1−q)}{m−1}, N ≥ 2 ; β < \frac{(1−q)}{m−1}, N = 1

Converse  yes  no  no

Non TCS property for

{exp(−1/∥x∥^{β}) \atop β > 2} ⋮ ⋮

Open questions

1. What happens for p = 2 and β = 2 ? It does not seem within sight.

2. We have no genuine converse for p > 2 and m > 1. A converse has been found for p = 2 because L^{2}(Ω) has an inner product. More precisely, for p > 2, ∫_{Ω} u^{p−1}v dx ≠ ∫_{Ω} v^{p−1}u dx in general. We search for another test-functions (see [3] for details).

3. When p > 2, we have a good generalization of the Cwikel, Lieb and Rosenblyum formula, that is, for large dimension (N > p). The estimate for N ≤ p is far from the optimum. When p = 2, the Lieb and Thirring formula works well. We hope that we will find an equivalent.

4. In [7], they also deal with second order elliptic equations with a strong absortion, i.e., u_{tt} + Δu − a(x)u^{q} = 0. Heuristically speaking, changing μ(α) into \sqrt{μ(α)} gives a sufficient condition for the TCS property. We are working on this type of equation when a depends also on t.

5. More generally, the following problem Δ_{p}u − a(x)u^{p−1} = 0 in an outside domain is difficult to handle. On ℝ^{N} minus a ball, a similar technique may be possible.

References

[1] Y. Belaud, Time vanishing properties for solutions of some degenerate parabolic equations with strong absorption, Advanced Nonlinear Studies 1, 2 (2001), 117-152.

[2] H. Brezis, Analyse fonctionnelle. Thorie et applications, Collection Mathmatiques appliques pour la matrise, Masson, 1986.

[3] Y. Belaud, B. Helffer, L. Vron, Long-time vanishing properties of solutions of sublinear parabolic equations and semi-classical limit of Schrdinger operator, Ann. Inst. Henri Poincarr Anal. nonlinear 18, 1 (2001), 43-68

[4] F.A. Berezin, M.A. Shubin, The Schrdinger Equation, Kluwer Academic Publishers, 1991.

[5] M. Cwikel, Weak type estimates for singular value and the number of bound states of Schrdinger operator, Ann. Math. 106 (1977), 93-100.

[6] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1977.

[7] V.A. Kondratiev and L. Vron, Asymptotic behaviour of solutions of some nonlinear parabolic or elliptic equations, Asymptotic Analysis 14 (1997), 117-156.

[8] B. Helffer, Semi-classical analysis for the Schrdinger operator and applications, Lecture Notes in Math. 1336, Springer-Verlag, 1989.

[9] E. H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schrdinger Hamiltonian and their relations to Sobolev Inequalities, In Studies in Math. Phys., essay in honour of V. Bargmann, Princeton Univ. Press, 1976.

[10] G. V. Rosenblyum, Distribution of the discrete spectrum of singular differential operators, Doklady Akad. Nauk USSR 202 (1972), 1012-1015.

[11] L. Véron, Effets régularisants de semi-groupes non linéaires dans des espaces de Banach, Annales faculté des Sciences Toulouse 1 (1979), 171-200.

[12] L. Véron, Coercivité et propriétés régularisantes des semi-groupes non linéaires dans les espaces de Banach, Publication de l’Université François Rabelais - Tours (1976).

Yves Belaud Laboratoire de Mathématiques et Physique Théorique, CNRS ESA 6083, Faculté des Sciences et Techniques, Université François Rabelais, 37200 Tours e-mail: belaud@univ-tours.fr