Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

Necessary Conditions.

For q>p-1, it is known (see [15, 26]) that in order for (1.1) to have a u with |∇⁡u|∈Llocq⁢(Ω) it is necessary that σ be regular and small enough. In particular, if σ is a signed measure, these necessary conditions can be quantified as

for all φ∈C0∞⁢(Ω). This can be seen by using |φ|qq-p+1 as a test function in (1.1) and applying the first inequality in (1.2) to get

Then by an appropriate Young’s inequality one arrives at (1.4) (see also [26] and [17]). Note that estimate (1.4) also holds when σ is a distribution in Wloc-1,qp-1⁢(Ω) in which case the left-hand side should be understood as 〈σ,|φ|qq-p+1〉.

Thus if σ is a nonnegative measure (or equivalently a nonnegative distribution) compactly supported in Ω, then condition (1.4) implies the capacitary condition

for every compact set K⊂Ω and a constant C independent of K. Here Cap1,s, s>1, is the capacity associated to the Sobolev space W1,s⁢(ℝn) defined for each compact set K⊂ℝn by

where χK is the characteristic function of K.

Moreover, in the case of nonnegative measure datum σ, all solutions of (1.1) must obey the regularity condition

for every compact set K⊂Ω. However, unlike (1.5), the constant C in (1.6) might depend on the distance from K to the boundary of Ω (see [15, 26]).

Motivated from (1.5), we now introduce the following definition.

It is well known that a measure μ∈M1,s⁢(Ω) if and only if the trace inequality

holds for all φ∈C0∞⁢(ℝn), with a constant C independent of φ. Here μ is extended by zero outside Ω. For this characterization see, e.g., [1]. Other characterizations are also available (see [20]).

In practice, it is useful to realize that the condition μ∈M1,s⁢(Ω) is satisfied if μ is a function verifying the Fefferman–Phong condition μ∈ℒ1+ϵ;s⁢(1+ϵ)⁢(Ω) for some ϵ>0 (see [10]). Here ℒ1+ϵ;s⁢(1+ϵ)⁢(Ω) is a Morrey space (see, e.g., [21]). In particular, it is satisfied provided μ is a function in the weak Lebesgue space Lns,∞⁢(Ω), s<n. Another sufficient condition is given by (G1*|μ|)ss-1∈ℒ1+ϵ;s⁢(1+ϵ)⁢(Ω) for some ϵ>0 (see [20]), where G1 is the Bessel kernel of order 1 defined via its Fourier transform by G1^⁢(ξ)=(1+|ξ|2)-12.

Now in view of (1.6), it is natural to look for a solution u of (1.1) such that |∇⁡u|q belongs to Mqq-p+1⁢(Ω). In this paper, we will be interested in only such a space of solutions.

Sufficient Conditions in Capacitary Terms.

There are many papers that obtain existence results for equation (1.1) under certain integrability conditions on the datum σ which are generally not sharp. The pioneering work [15] originally used capacities to treat (1.1) in the ‘linear’ case p=2 in ℝn (q>1), or in a bounded domain Ω (q>2). For p>2-1n and q≥1, it was shown in [28, 30] (see also [13, 27] for the sub-critical case p-1<q<n⁢(p-1)n-1) that, under certain regularity conditions on 𝒜 and ∂⁡Ω, if σ is a finite signed measure in Mqq-p+1⁢(Ω), with [σ]Mqq-p+1⁢(Ω) being sufficiently small, then equation (1.1) admits a solution u∈W01,q⁢(Ω) such that |∇⁡u|q∈Mqq-p+1⁢(Ω). Similar existence results have recently been extended to the case 3⁢n-22⁢n-1<p≤2-1n, q≥1, in [23] and to the case 1<p≤3⁢n-22⁢n-1, q≥1, in [25]. We also mention that the earlier work [26, 29] covers all p>1 but only for q>p.

We observe that whereas the existence results of [15, 28, 23, 25, 26] are sharp when σ is a nonnegative measure, they could not be applied to a large class distributional data σ with strong oscillation. Take for example the function

where s=qq-p+1 and ϵ>0 such that ϵ+s<n. Then σ=|f⁢(x)|⁢d⁢x fails to satisfy the capacitary inequality (1.5), but it is possible to show that the equation

admits a solution u∈W01,q⁢(B1⁢(0)) provided |λ| is sufficiently small. For this see [21] which addresses oscillatory data in the Morrey space framework. See also [2, 5, 11, 12] in which the case q=p is considered. Note that in this special case, the Riccati-type equation -div⁡𝒜⁢(x,∇⁡u)=|∇⁡u|p+σ is strongly related to the Schrödinger-type equation -div⁡𝒜⁢(x,∇⁡u)=σ⁢|u|p-2⁢u (see [14]). This relation has been employed in an essential way in [16, 17] to study the existence of local solutions in this case. Here by a local solution we mean one that belongs to Wloc1,p⁢(Ω) and has no pre-specified boundary condition.

Main Results.

The first main result of this paper is to treat (1.1) with oscillatory data in the framework of the natural space M1,qq-p+1⁢(Ω). This provides non-trivial improvements of the results of [15, 28, 23, 25, 26] and [21] at least in the case q>p. We first observe the following necessary condition on σ so that (1.1) has a solution u such that |∇⁡u|q∈M1,qq-p+1⁢(Ω).

Conversely, when q>p, we obtain the following existence result.

The notion of (δ,R0)-Reifenberg flat domains mentioned in Theorem 1.3 is made precise by the following definition.

Examples of such domains include those with C1 boundaries or Lipschitz domains with sufficiently small Lipschitz constants. They also include certain domains with fractal boundaries.

On the other hand, the (δ,R0)-BMO condition imposed on 𝒜⁢(x,ξ) allows it to have small jump discontinuities in the x-variable. More precisely, given two positive numbers δ and R0, we say that 𝒜⁢(x,ξ) satisfies the (δ,R0)-BMO condition if

where for each ball B=Br⁢(y) we let

with

Thus one can think of the (δ,R0)-BMO condition as an appropriate substitute for the Sarason VMO condition.

The second main result of the paper is to treat (1.1) for the case p>3⁢n-22⁢n-1, p-1<q<1, and σ is a signed measure compactly supported in Ω. This extends the results of [23] to the sublinear range p-1<q<1, which cannot be dealt with by the method of [23] due to the lack of convexity. However, here we assume that 𝒜⁢(x,ξ) is Hölder continuous in the x-variable, i.e.,

for some θ∈(0,1) and all x,x0,ξ∈ℝn. We note that this regularity assumption can be relaxed by using a weaker Dini’s condition as in [24]. Moreover, for Ω we further assume the following integrability condition (besides the (δ,R0)-Reifenberg flatness condition):

for some ϵ0>0. Here d⁢(x) is the distance from x to ∂⁡Ω, i.e., d⁢(x)=inf⁡{|x-y|:y∈∂⁡Ω}. It is not clear to us if the (δ,R0)-Reifenberg flatness condition for a sufficiently small δ will imply (1.12). Note that (1.12) holds (even with any 0<ϵ0<1) for any bounded Lipschitz domain. More generally, (1.12) holds for some ϵ0>0 provided we can find an ϵ>0 such that

holds for all small τ>0.

We refer to [9] for the notion of renormalized solutions. Note that in the case p≤2-1n the gradients of such solutions should be interpreted appropriately.