The aim of this paper is to extend the usual framework of PDE with Au=-divax,u,∇u to include a large class of cases with Au=∑β≤α-1βDβAβx,u,∇u,…,∇αu, whose coefficient Aβ satisfies conditions (including growth conditions) which guarantee the solvability of the problem Au=f. This new framework is conceptually more involved than the classical one includes many more fundamental examples. Thus our main result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation.
1. Introduction
This paper is motivated by the study of the unilateral problem associated with the following equation:
(1)A(u)+g(x,u)=f.
We show the existence of variational solutions of this elliptic boundary value problem for strongly elliptic systems of order 2m on a domain Ω in ℝN in generalized divergence form as follows:
(2)A(u)=∑|β|≤α(-1)|β|DβAβ(u,∇u,…,∇αu).
The function g satisfies a sign condition but has otherwise completely unrestricted growth with respect to u.
Equations of type (1) were first considered by Browder [1] as an application to the theory of not everywhere defined mapping of monotone type. For α=1, that is, A of second order, their solvability under fairly general and natural assumptions was proved by Hess [2]. The treatment of the case α>1 is more involved due to the lack of a simple truncation operator in higher order Sobolev spaces. Webb [3] observed that rather delicate approximation procedure introduced in nonlinear potential theory by Hedberg [4] could be used in place of truncation. This yielded the solvability of (1) for α>1. Brezis and Browder [5] then used this approximation procedure to solve a question which they had considered earlier [6] about the action of some distribution. They also showed that their result on the action of some distributions could itself be used in place of truncation in the study the problem (1). In a more general case, Boccardo et al. studied inequations associated with (1), see [7].
The functional setting in all the results mentioned above is that of the usual Sobolev spaces Wα,p(ℝN), and the functions Aβ in (2) are supposed to satisfy polynomial growth conditions with respect to u and its derivatives. Benkirane and Gossez established this result in the Orlicz-Sobolev spaces WαLA(ℝN), see [8–10].
It is our purpose in this paper to study these problems in this setting of Sobolev spaces with variable exponent Wα,p(·)(ℝN) of the harder higher order case α>1. We consider problem (1) as well as Hedberg’s approximation theorem and Brezis-Browder’s question on the action of some distributions.
The paper is structured as follows. After some necessary preliminaries, in Section 3, we give the proof of the approximation theorem. In addition, Section 4 forms a useful supplement to some applications of (1).
2. Preliminaries
In this section we list briefly some definitions and well known facts about Sobolev spaces with variable exponent and Bessel potential spaces with variable exponent. Standard references are [11, 12].
Let Ω be an open subset of ℝN, by the symbol 𝒫(Ω), we denote the family of all measurable functions p(·):Ω→[1,∞].
For p(·)∈𝒫(Ω), put
(3)p+=esssupx∈Ωp(x),p-=essinfx∈Ωp(x).
Furthermore, we introduce a class ℬ(Ω) by
(4)ℬ(Ω):={p∈𝒫(Ω);1<p-≤p+<∞}.
Let p(·)∈ℬ(Ω), and consider the functional
(5)ϱp(·)(f)=∫Ω|f(x)|p(x)dx,
on all measurable function f on Ω. The Lebesgue space with variable exponent Lp(·)(Ω) is defined as the set of all measurable functions f on Ω such that, for some λ>0,
(6)ϱp(·)(fλ)<∞,
equipped with the norm
(7)∥f∥p(·)=inf{λ>0;ϱp(·)(fλ)≤1}.
The space (Lp(·)(Ω),∥·∥p(·)) is a separable Banach space. Moreover, if 1<p-≤p+<+∞, then Lp(·)(Ω) is uniformly convex, hence, reflexive, and its dual space is isomorphic to Lp′(·)(Ω), where 1/p(x)+1/p′(x)=1.
Finally, we have the Hölder type inequality as follows:
(8)|∫Ωuvdx|≤(1p-+1p-′)∥u∥p(·)∥v∥p′(·)
for all u∈Lp(·)(Ω) and v∈Lp′(·)(Ω).
The Hardy-Little-wood maximal operator M is defined on locally integrable functions f on Ω by the following formula:
(9)Mf(x)=supR>01|B(x,R)|∫B(x,R)|f(y)|dy,
where B(x,R) denote the open ball in Ω with center x and radius R, and |B(x,R)| denotes the volume of B(x,R).
Definition 1.
By ℳ(Ω), denote the class of all functions p∈ℬ(Ω) for which the operator M is bounded on Lp(·)(Ω); that is,
(10)∥Mf∥p(·)≤C∥f∥p(·),
with a positive constant C independent of f.
Remark 2.
For example, p(·)∈ℳ(Ω) if the following two conditions are satisfied:
(11)|p(x)-p(y)|≤c-log(|x-y|),|x-y|≤12,|p(x)-p(y)|≤clog(e+|x|),|y|>|x|.
For more details, see [13–16], where various sufficient conditions for p(·)∈ℳ(Ω) can be found.
Let p(·)∈ℬ(Ω) and α∈ℕ; we define the Sobolev space with variable exponent by
(12)Wα,p(·)(Ω)={u,Dβu∈Lp(·)(Ω)if|β|≤α},
equipped with the norm
(13)∥u∥Wα,p(·)=∑|β|≤α∥Dβu∥Lp(·),
where β∈ℕ0N is a multi-index, |β|=β1+⋯+βN, and Dβ=∂|β|/∂β1x1⋯∂βNxN.
Next, we define W0α,p(·)(Ω) as the closure of 𝒟(Ω) in Wα,p(·)(Ω) and W-α,p′(·)(Ω) the dual space of Wα,p(·)(Ω), where 1/p(·)+1/p′(·)=1.
The Bessel kernel Gα (see [17]) of order α>0 is defined by
(14)Gα(x)=πN/2Γ(α/2)∫0∞e-s-π2|x|2/ss(α-N)/2dss,gggggggggggggggggggggggx∈ℝN.
The Riesz kernel Iα (see [18]) of order α>0 is defined by
(15)Iα(x)=γα|x|N-α,
where γα is a certain constant, whose exact value is
(16)γα=Γ((N-α)/2)(πN/22αΓ(α/2)).
It follows easily that
(17)Gα(x)~Iα(x),|x|⟶0,0<α<N,
and an examination of (14) shows without much effort that for any c<1,
(18)Gα(x)=o(e-c|x|),|x|⟶∞,N>α>0,∫|x-y|<δf(y)dy|x-y|N-α≤AδαMf(x).
Writing Gα(x)=Gα(r), r=|x|, this implies that, for N>α>1,
(19)Gα′(r)~-(N-α)Gα-1(r),r⟶0,N>α>0,Gα′(r)~-aαr(α-N-1)/2e-r~-cGα(r),r⟶∞,
with c=2(α+N-3)/2π(N-α-1)/2 and 1/aα=(4π)(α)/2Γ((α)/2).
3. Main Results
First, we give the following results which will be used in our main result.
3.1. Useful Results
Let p(·)∈ℬ(ℝN) and α>0. The Bessel potential space with variable exponent Lα,p(·)(ℝN) is defined, for N>α>0, by
(20)Lα,p(·)(ℝN)={u=Gα*f;f∈Lp(·)(ℝN)},
and is equipped with the norm
(21)∥u∥α,p(·)=∥f∥p(·).
Lemma 3 (see [12]).
If p(·)∈ℳ(ℝN) and α∈ℕ, then
(22)Lα,p(·)(ℝN)≅Wα,p(·)(ℝN),
and the corresponding norms are equivalent.
Lemma 4 (see [12]).
Suppose that p(·)∈ℳ(ℝN) and α≥0. Then there exists a positive constant C such that
(23)∥Gα*f∥p(·)≤C∥f∥p(·),forf∈Lp(·)(ℝN).
We will now verify that un satisfies all the required properties in Proposition 7. The argument relies heavily on the following Lemmas 5 and 6. In the sequel, we need the following two technical lemmas.
Lemma 5.
If 1<p-≤p(x)≤p+<∞ and for any multi-index |ξ|<α<N, there exists a constant A such that for any f∈Lp(·)(ℝN),
(24)|Dξ(Gα*f(x))|≤A(Mf(x))|ξ|/α(Gα*|f|(x))1-|ξ|/α,foralmosteveryx∈ℝN.
Proof.
We assume that Mf(x)<∞; otherwise, there is nothing to prove. We then observe that there is a constant A1 such that Gα*|f|(x)≤A1Mf(x). In fact, by (17), (18), and ([11], Lemma 6.1.4), there exists a constant A such that(25)Gα*|f|(x)≤A∫|x-y|<1|f(y)|dy|x-y|N-α+A∫|x-y|≥1|f(y)|e-|x-y|/2dy≤AMf(x)+A∑i=1∞e-i/2∫i+1>|x-y|≥i|f(y)|dy≤AMf(x)+AMf(x)∑i=1∞(i+1)Ne-i/2=A1Mf(x).
Then, by (19) and (18), for any δ≤1, we have
(26)|Dξ(Gα*|f|(x))|≤A∫|x-y|<δ|f(y)|dy|x-y|N-α+|ξ|+A∫δ≤|x-y|<1|f(y)|dy|x-y|N-α+|ξ|+A∫|x-y|≥1Gα(x-y)|f(y)|dy≤A(δα-|ξ|Mf(x)+δ-|ξ|(Gα*|f|(x))+(Gα*|f|(x))δα-|ξ|Mf(x)).
Now choosing δα=Gα*|f|(x)/A1Mf(x); then, δ≤1 and the result follows.
Lemma 6.
Suppose that 1<p-≤p(x)≤p+<∞ and 0<α<N. Let H∈Ck(ℝ+) for some k≥α, if H satisfies
(27)supt>0|ti-1H(i-1)(t)|≤L<∞,withi=1,2,…,k.
Then H(Gα*f)∈Lα,p(·)(ℝN) for every f∈L+p(·)(ℝN), and there exists a constant A, depending only on α and N, such that
(28)∥H(Gα*f)∥α,p(·)≤AL∥Gα*f∥α,p(·)=AL∥f∥p(·).
Proof.
For α integer. Assume that f∈C0∞(ℝN) and f≥0. Set u=Gα*f and notice that u(x)>0 for all x, so that H(u) is defined. If ξ is a multi-indix with |ξ|=α, we find by the chain rule that
(29)Dξ(H(u))=∑i=1αH(i)(u)∑cξDξ1u⋯Dξiu,
where the interior sum is over all ordered i-tuples of multi-indices {ξ1,…,ξi} such that ξ1+⋯+ξi=ξ, and all |ξj|≥1. The cξ are coefficients, whose exact value is of no consequence to us. Thus, by assumption of Lemma 3.4 we get
(30)|Dξ(H(u))|≤AL∑i=1αu1-i∑|Dξ1u⋯Dξiu|.
For i>1, we estimate these derivatives by means of Lemma 5. By the positivity of f, we have
(31)|Dξju|≤A(Mf(x))|ξj|/αu1-|ξj|/α.
Thus, since ∑j=1i(1-|ξj|/α)=i-|ξ|/α=i-1,
(32)∑i=2αu1-i∑|Dξ1u⋯Dξiu|≤A∑i=2αu1-iMfui-1=AMf.
Taking the term with i=1 into account, we obtain
(33)|Dξ(H(u))|≤AL(|Mf|+|Dξu|).
But we already know from (10) that ∥Mf∥p(·)≤A∥f∥p(·) and that ∥Dξ(Gα*f)∥p(·)≤A∥f∥p(·) for |ξ|=α.
This finishes the proof for smooth f.
Now we pass to the general case and let f be an arbitrary function in L+p(·)(ℝN). Then there are nonnegative functions fi∈C0∞(ℝN), i=1,2,…, such that
(34)limi→∞∥fi-f∥p(·)=0.
By the first part of the proof,
(35)∥H((Gα*fi))∥α,p(·)≤AL∥f∥p(·),
for all sufficiently large i.
Thus, setting H((Gα*fi))=Gα*gi, we can assume that {gi}1∞ converges weakly in Lp(·)(ℝN) to an element g, with ∥g∥p(·)≤AL∥f∥p(·).
We have to prove that Gα*g=H((Gα*f)).
The strong convergence of {fi}1∞ and the fact that Gα∈L1(ℝN) imply, by Lemma 4, that {Gα*fi}1∞ converges strongly in Lp(·)(ℝN) to Gα*f. After extraction of a subsequence, we can assume that
(36)limi→∞Gα*fi(x)=Gα*f(x)a.e.
But H is continuous, so it follows that
(37)limi→∞Gα*gi(x)=limi→∞H((Gα*fi))(x)=H((Gα*f))(x)a.e.
On the other hand, the weak convergence of {gi}1∞ implies that the pointwise limit of {Gα*gi}1∞ (which is now known to exist a.e.) is Gα*g. In fact, setting gi-g=hi, for an arbitrary ϵ>0,
(38)Gα*hi(x)=∫|x-y|≤ϵGα(x-y)hi(y)dy+∫|x-y|>ϵGα(x-y)hi(y)dy.
By weak convergence, the last term tends to zero, since Gα is in Lp′(·)(ℝN) away from the origin. Consider the following:
(39)Letλ>liminfi→∞∥∫|y|≤ϵGα(y)hi(·-y)dy∥p(·).
We deduce that, for all j, λ>infi≥j∥∫|y|≤ϵGα(y)hi(·-y)dy∥p(·), which implies that for all j, there exists i≥j, such that
(40)λ>∥∫|y|≤ϵGα(y)hi(·-y)dy∥p(·).
Then,
(41)∫|y|≤ϵ(liminfi→∞Gα(y)hi(x-y)λ)p(x)dx≤liminfi→∞∫|y|≤ϵ(Gα(y)hi(·-y)λ)p(x)dx≤1.
Since λ≥∥liminfi→∞Gα*hi∥p(·), then we have
(42)∥limi→∞Gα*hi∥p(·)≤liminfi→∞∥∫|y|≤ϵGα(y)hi(·-y)dy∥p(·)≤supi∥hi∥p(·)∫|y|≤ϵGα(y)dy,
which is an arbitrary small number, and thus for a.e. x,
(43)Gα*g(x)=limi→∞Gα*gi(x)=limi→∞H((Gα*fi))(x)=H((Gα*f))(x).
This completes the proof of Lemma 5.
Proposition 7.
Let u∈Wα,p(·)(ℝN), there exist a sequence un, such that
un∈Wα,p(·)(ℝN)∩L∞(ℝN),suppun is compact;
|un(x)|≤|u(x)| and un(x)u(x)≥0 a.e. in ℝN;
un→u in Wα,p(·)(ℝN) as n→∞.
Proof.
The proof of Proposition 7 is done in two steps as follows.
Step1(caseαp^->N). Let ξ∈C0∞(ℝN) be a fixed function such that 0≤ξ≤ 1 and ξ(x)=1 in a neighborhood of the origin.
Let ξn(x) = ξ(x/n); then, un(x) = ξn(x)u(x) satisfies all the required properties, using the fact that u∈L∞(Ω) by Sobolev’s theorem [19].
Step2 (caseαp-≤N). We assume that u has compact support, if necessary by multiplying with a suitable ξn. We represent u as a Bessel potential, u=Gα*f, so that ∥f∥p(·)≤A∥u∥Wα;p(·). Set
(44)v=Gα*|f|,
and let T∈C∞(ℝ) be a function such that 0≤T≤1, T(t)=1 for 0≤t≤1/2 and T(t)=0 for t≥1. Then, set
(45)un(x)=T(v(x)n)u(x),n=1,2,3,….
We First observe that, un(x)=0 on the set {x:v(x)≥n}, which includes {x:|u(x)|≥n}, and so we have |un(x)|<n a.e., and thus un(x)u(x)≥0. It remains to prove that un∈Wα,p(·)(ℝN) and that un converges to u as n tends to ∞.
Let η be any multi-index with 0<|η|=β≤α. If η = η1+⋯+ηi, i>1, and all |ηj|≥1, we find by the same arguments as in the proof of Lemma 6 that
(46)|DηT(v(x)n)|≤A∑i=1βn-i∑|Dη1v(x)⋯Dηiv(x)|.
By Lemma 5, we have, for any multi-index η with 0<|η|<α,
(47)|Dηv(x)|≤AMf(x)|η|/α(v(x))1-|η|/α.
On the open set {x:v(x)>n}, using the fact that DηT(v(x)/n)=0; then,
(48)|DηH(v(x)n)|≤A∑i=1βn-ini-|η|/αMf(x)|η|/α≤An-|η|/αMf(x)|η|/α,
for |η|≤α-1, and
(49)|DηT(v(x)n)|≤An-1(Mf(x)+|Dηv(x)|),
for |η| = α.
By using Leibniz’s formula, we have for |ξ| = α, if v(x)≤n, again using Lemma 5, that
(50)|Dξun(x)-Dξu(x)|=|(1-T(v(x)n))Dξu(x)|,|Dξun(x)-Dξu(x)|≤A∑0<|η|<α|DηT(v(x)n)||Dξ-ηu(x)|+|DξH(v(x)n)||u(x)|≤A∑0<β<αn-β/αMf(x)β/αMf(x)1-β/αv(x)β/α+An-1(Mf(x)+|Dξv(x)|)v(x)≤A(v(x)n)1/αMf(x)+A(v(x)n)(Mf(x)+|Dξv(x)|).
If v(x)>n, we have Dξun(x)=0. It follows that a.e.
(51)limn→∞|Dξun(x)-Dξu(x)|=0,|Dξun(x)-Dξu(x)|≤|Dξu(x)|+A(Mf(x)+|Dξv(x)|).
The functions on the right hand side belong to Lp(·), so the theorem follows by applying the dominated convergence.
Remark 8.
The sequence un constructed above satisfies
(52)∥un∥Wα,p(·)≤C∥u∥Wα,p(·),
with a constant C depending only on α and N.
3.2. Existence Result
This subsection is devoted to establish the following existence theorem.
Theorem 9.
Let T∈Lloc1(ℝN)∩W-α,p′(·)(ℝN) and u∈Wα,p(·)(ℝN). Assume that T(x)u(x)≥h(x) a.e. in ℝN, for some h∈L1(ℝN). Then,
(53)Tu∈L1(ℝN),∫ℝNT(x)u(x)dx=〈T,u〉.
Proof.
We first deduce Theorem 9 as a simple consequence of Proposition 7. Let un be a sequence defined in Proposition 7. It follows easily from (i) in Proposition 7 (using convolution with mollifiers and according to [11]) that
(54)∫ℝNT(x)un(x)dx=〈T,un〉.
By Proposition 7, the right hand side of (54) converges as n→∞ to 〈T,u〉. On the other hand, we have Tun≥-|h| a.e. We deduce from Fatou’s lemma that Tu∈L1. We conclude by dominated convergence that
(55)∫T(x)un(x)dx⟶∫T(x)u(x)dxandthus∫T(x)u(x)dx=〈T,u〉.
Theorem 10.
Let T∈Lloc1(Ω)∩W-α,p′(·)(Ω) be such that
(56)∫Ω∩B(0,R)|T(x)|dx<∞foreveryR<∞.
Assume that u∈W0α,p(·)(Ω) and T(x)u(x)≥h(x) a.e. in Ω, for some h∈L1(Ω). Then,
(57)Tu∈L1(Ω),∫ΩT(x)u(x)dx=〈T,u〉.
Proof.
The proof is straightforward when αp->N; therefore, we may assume that αp-≤N. Using ξnu in place of u, we may always reduce to the case where suppu is bounded. Set(58)u¯={u(x)ifx∈Ω,0ifx∈ℝN∖Ω.
Then u¯∈Wα,p(·)(ℝN). This allows us to write
(59)u¯=Gα*f,
for some f in Lp(·)(ℝN). As in the proof of Proposition 7, set
(60)v=Gα*|f|,un=H(vn)u.
Since u∈W0α,p(·)(Ω), there exists a sequence uj∈C0∞(Ω) such that uj→u in Wα,p(·)(Ω) and a.e. (see [19]). For each j, we perform the above construction and we set
(61)u¯j=Gα*fj,vj=Gα*|fj|,unj=H(vjn)uj.
Fix ξ∈C0∞(ℝN). We clearly have
(62)∫ΩTξunjdx=〈T,ξunj〉.
As we keep n fixed and let j→∞, we see that
(63)∫ΩTξunjdx⟶∫ΩTξundx,
by dominated convergence and (56).
On the other hand, by Remark 8, we obtain
(64)∥unj∥Wα,p(·)(Ω)≤∥u¯j∥Wα,p(·)(ℝN)≤C,
where C does not depend on j and n.
Therefore, unj converges weakly to un in W0α,p(·)(Ω) as j→∞ and thus ξunj⇀ξun as j→∞. Passing to the limit in (62) as j→∞, we find
(65)∫ΩTξun=〈T,ξun〉.
We conclude easily (by the argument as in the proof of Theorem 9) that
(66)Tu∈L1(Ω),∫ΩT(x)u(x)dx=〈T,u〉.
4. An Application to a Strongly Nonlinear Elliptic Equation
Let Ω be an open set on ℝN, and assume that A:W0α,p(·)(Ω)→W-α,p′(·)(Ω) is a pseudomonotone operator which maps bounded sets into bounded sets and which is coercive. And let g(x,s):Ω×ℝ→ℝ be a Carathéodory function satisfying the sign condition (Ω×ℝ) and for each t>0, there exists ht∈L1(Ω) as follows:
(67)sup|s|≤t|g(x,s)|≤ht(x).
Theorem 11.
For every f∈W-α,p′(·)(Ω), there exists u∈W0α,p(·)(Ω) such that
(68)g(x,u)∈L1(Ω),g(x,u)u∈L1(Ω),〈Au,v〉+∫Ωg(x,u)vdx=〈f,v〉,∀v∈W0α,p(·)(Ω)∩L∞(Ω).
Furthermore, if g is nondecreasing in u and if u1and u2 are two solutions corresponding to f1 and f2, respectively, then
(69)〈Au1-Au2,u1-u2〉+∫Ω[g(x,u1)-g(x,u2)](u1-u2)dx=〈f1-f2,u1-u2〉.
Proof.
Let
(70)gn(x,s)={η(xn)g(x,s)if|g(x,s)|≤nη(xn)nsign(g(x,s))if|g(x,s)|>n,
where η∈𝒟(ℝN) with 0≤η≤1 and η(x)=1 near x=0. It follows easily from the theory of pseudomonotone operators that there exists un∈W0α,p(·)(Ω) such that
(71)Aun+gn(x,un)=f.
In addition,
(72)∥un∥W0α,p(·)(Ω)≤C,∫Ωgn(x,un)un≤C.
Then we can assume that
(73)un⇀uweaklyinW0α,p(·)(Ω),a.einΩ,Aun⇀χweaklyinW-α,p′(·)(Ω).
Moreover,
(74)|gn(x,un)|≤sup|t|≤s|g(x,t)|+1sgn(x,un)un,
which implies that, for any measurable subset E⊂Ω,
(75)∫E|gn(x,un)|≤∫Ehs(t)+Cs.
Then, thanks to Vitali’s theorem, we deduce that
(76)gn(x,un)⟶g(x,u).
By Fatou’s lemma, it is easy to see that
(77)limsupn〈Aun,un〉≤〈f,u〉-∫Ωg(x,u)dx.
Set T=g(x,u)=f-χ, we have T∈L1(Ω)∩W-α,p′(·)(Ω) and as a consequence of Theorem 10, we conclude that
(78)∫Ωg(x,u)udx=〈f-χ,u〉.
Therefore,
(79)limsupn〈Aun,un〉≤〈χ,u〉andconsequentlyAu=χ.
The conclusion follows readily. (68) is again a direct consequence of Theorem 11.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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