# Solution of Nonlinear Elliptic Boundary Value Problems

Authors: Li Wei; Liling Duan; Haiyun Zhou
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study a kind of nonlinear elliptic boundary value problems with generalizedp-Laplacian operator. The unique solution is proved to be existing and the relationship between this
solution and the zero point of a suitably defined nonlinear maximal monotone operator is investigated.
Moreover, an iterative scheme is constructed to be strongly convergent to the unique solution. The
work done in this paper is meaningful since it combines the knowledge of ranges for nonlinear operators,
zero point of nonlinear operators, iterative schemes, and boundary value problems together. Some new
techniques of constructing appropriate operators and decomposing the equations are employed, which
extend and complement some of the previous work.


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## Body

## 1. Introduction

The study on nonlinear boundary value problems withp-Laplacian operator, Δp, is a hot topic since it has a close relationship with practical problems. Some significant work has been done by us, see [1–8], and so forth.Specifically, in 2004, we studied the following nonlinear elliptic boundary value problem involving the generalizedp-Laplacian operator:
(1.1)-div[(C(x)+|∇u|2)(p-2)/2∇u]+|u|p-2u+g(x,u(x))=f(x),a.e.inΩ-〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉=0,a.e.onΓ.In Wei and Hou [4], we proved that under some conditions (1.1) has solutions in L2(Ω), where 2≤p<+∞. In [5, 6], we extended our work to the following problem:
(1.2)-div[(C(x)+|∇u|2)(p-2)/2∇u]+|u|p-2u+g(x,u(x))=f(x),a.e.inΩ-〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉∈βx(u(x)),a.e.onΓ.
In Wei and Zhou [5], we established that (1.2) has solutions in Lp(Ω), where 2≤p<+∞, and in Wei [6] we proved that (1.2) has solutions in Ls(Ω), where max(N,2)≤p≤s<+∞. As the summary and extension of [5, 6], we studied the following nonlinear boundary value problem:
(1.3)-div[(C(x)+|∇u|2)(p-2)/2∇u]+ε|u|q-2u+g(x,u(x))=f(x),a.e.inΩ-〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉∈βx(u(x)),a.e.onΓ.It was shown by Wei and Agarwal [7] that (1.3) had a solution in Ls(Ω), where 2N/(N+1)<p≤s<+∞, 1≤q<+∞ if p≥N and 1≤q≤Np/(N-p) if p<N, for N≥1.Clearly, ifC(x)≡0, then (1.1), (1.2), and (1.3) reduce the cases of involving p-Laplacian operators.It is worth to mention that all of the work done in [4–7] is based on a perturbation result of the ranges of m-accretive mappings by Calvert and Gupta [9].In this paper, we will study the following nonlinear elliptic boundary value problem:(1.4)-div[(C(x)+|∇u|2)(p-2)/2∇u]+ε|u|q-2u+g(x,u(x),∇u(x))=f(x),a.e.inΩ-〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉∈βx(u(x)),a.e.onΓ.Necessary details of (1.4) will be provided in Section 3.We may notice that the principal part of the concerned equation is almost the same as those in (1.1), (1.2), and (1.3) while the nonlinear item g(x,u(x)) is replaced by the item g(x,u(x),∇u(x)), which is rather general. It seems that the difference is minor; however, the previous method cannot be employed. In this paper, we will use some perturbation results of the ranges for maximal monotone operators by Pascali and Sburlan [10] to prove that (1.4) has a unique solution in W1,p(Ω) and later we will prove that the unique solution is the zero point of a suitably defined maximal monotone operator. Finally, we will employ an iterative scheme to approximate strongly to the unique solution. Some new ideas of combining the knowledge of the ranges of nonlinear operators, zero point of nonlinear operators, iterative schemes, and the solution of nonlinear boundary value problems are demonstrated in this paper.


## 2. Preliminaries

Now, we list some of the knowledge we need in the sequel.LetX be a real Banach space with a strictly convex dual space X*. We use (·,·) to denote the generalized duality pairing between X and X*. We use “→” to denote strong convergence. Let “X↪Y” denote the space X embedded continuously in space Y. For any subset G of X, we denote by intG its interior.FunctionΦ is called a proper convex function on X [11] if Φ is defined from X to (-∞,+∞], not identically +∞ such that Φ((1-λ)x+λy)≤(1-λ)Φ(x)+λΦ(y), whenever x,y∈X and 0≤λ≤1.FunctionΦ:X→(-∞,+∞] is said to be lower semicontinuous on X [11] if liminfy→xΦ(y)≥Φ(x), for any x∈X.Given a proper convex functionΦ on X and a point x∈X, we denote by ∂Φ(x) the set of all x*∈X* such that Φ(x)≤Φ(y)+(x-y,x*), for every y∈X. Such elements x* are called subgradients of Φ at x, and ∂Φ(x) is called the subdifferential of Φ at x [11].A single-valued mappingT:D(T)=X→X* is said to be hemicontinuous [11] if w-limt→0T(x+ty)=Tx, for any x,y∈X.A multivalued mappingA:X→2X* is said to be monotone [10] if its graph G(A) is a monotone subset of X×X* in the sense that
(2.1)(u1-u2,w1-w2)≥0,
for any [ui,wi]∈G(A),  i=1,2. The mapping A is said to be strictly monotone if the equality in (2.1) implies that u1=u2. The monotone operator A is said to be maximal monotone if G(A) is maximal among all monotone subsets of X×X* in the sense of inclusion. The mapping A is said to be coercive [10] if limn→+∞(xn,xn*)/∥xn∥=+∞ for all [xn,xn*]∈G(A) such that limn→+∞∥xn∥=+∞. A point x∈D(A) is said to be a zero point of A if 0∈Ax, and we denote by A-1(0)={x∈X:0∈Ax} the set of zero points of A.Lemma 2.1 (Adams [12]).
LetΩ be a bounded conical domain in RN. If mp>N, then Wm,p(Ω)↪CB(Ω); if mp<N and q=Np/(N-mp), then Wm,p(Ω)↪Lq(Ω); if mp=N and p>1, then for 1≤q<+∞,Wm,p(Ω)↪Lq(Ω).Lemma 2.2 (Pascali and Sburlan [10]).
IfB:X→2X* is an everywhere defined, monotone, and hemicontinuous operator, then B is maximal monotone.Lemma 2.3 (Pascali and Sburlan [10]).
IfΦ:X→(-∞,+∞] is a proper convex and lower semicontinuous function, then ∂Φ is maximal monotone from X to X*.Lemma 2.4 (Pascali and Sburlan [10]).
IfB1 and B2 are two maximal monotone operators in X such that (intD(B1))⋂‍D(B2)≠∅, then B1+B2 is maximal monotone.Lemma 2.5 (Pascali and Sburlan [10]).
IfA:X→2X* is maximal monotone and coercive, then R(A)=X*.Definition 2.6 (Kamimura and Takahashi [13]).
LetX be a real smooth Banach space. Then the Lyapunov functionalφ:X×X→R+ is defined as follows:
(2.2)φ(x,y)=∥x∥2-2(x,Jy)+∥y∥2,∀x,y∈X,
where J:X→2X* is the duality mapping defined by Jx={f∈X*:(x,f)=∥x∥∥f∥,∥f∥=∥x∥}, for x∈X.Lemma 2.7 (Kamimura and Takahashi [13]).
LetX be a real reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed and convex subset of X, and let x∈X. Then there exists a unique element x0∈C such that
(2.3)φ(x0,x)=min{φ(z,x):z∈C}.Define a mappingΠC from X onto C by ΠCx=x0 for all x∈X. Then ΠC is called the generalized projection mapping from X onto C. It is easy to see that ΠC coincides with the metric projection PC in a Hilbert space.


## 3. Main Results

### 3.1. Notations and Assumptions of (1.4)

In the following of this paper, unless otherwise stated, we will assume that2N/(N+1)<p<+∞,  1≤q<+∞ if p≥N, and 1≤q≤Np/(N-p) if p<N, for N≥1. Let 1/p+1/p′=1. We use ∥·∥p,∥·∥p', and ∥·∥1,p,Ω to denote the norm of spaces Lp(Ω),Lp′(Ω), and W1,p(Ω), respectively.In nonlinear boundary value problem (1.4), Ω is a bounded conical domain of an Euclidean space RN with its boundary Γ∈C1 (see Wei and He [1]). We will assume that Green’s formula is available. f(x)∈Lp′(Ω) is a given function. 0≤C(x)∈Lp(Ω),  ε is a nonnegative constant and ϑ denotes the exterior normal derivative of Γ.Letφ:Γ×R→R be a given function such that, for each x∈Γ,  φx=φ(x,·):R→R is a proper, convex, and lower semicontinuous function with φx(0)=0. Let βx be the subdifferential of φx, that is, βx≡∂φx. Suppose that 0∈βx(0) and for each t∈R, the function x∈Γ→(I+λβx)-1(t)∈R is measurable for λ>0.Suppose thatg:Ω×RN+1→R is a given function satisfying the following conditions, which can be found in Zeidler [14].(a)
Carathéodory's conditions:(3.1)x⟶g(x,r)ismeasurableonΩ∀r∈RN+1,r⟶g(x,r)iscontinuousonRN+1foralmostallx∈Ω.(b)
Growth condition:(3.2)g(x,r1,…,rN+1)≤h1(x)+b∑i=1N+1|ri|p-1,
where (r1,r2,…,rN+1)∈RN+1,  h1(x)∈Lp(Ω) and b is a fixed positive constant.(c)
Monotone condition:g is monotone with respect to r1, that is,
(3.3)(g(x,r1,…,rN+1)-g(x,t1,…,tN+1))(r1-t1)≥0,
for all x∈Ω and (r1,…,rN+1),(t1,…,tN+1)∈RN+1.


### 3.2. Existence and Uniqueness of the Solution of (1.4)

Lemma 3.1.
Define the mappingBp,q:W1,p(Ω)→(W1,p(Ω))* by
(3.4)(v,Bp,qu)=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇v〉dx+ε∫Ω‍|u(x)|q-2u(x)v(x)dx
for any u,v∈W1,p(Ω). Then, Bp,q is everywhere defined, strictly monotone, hemicontinuous, and coercive.Moreover, Lemma2.2 implies that Bp,q is maximal monotone.Proof.
From Lemma  3.2 by Wei and Agarwal [7], we know that Bp,q is everywhere defined, monotone, hemicontinuous, and coercive. Then we only need to show that Bp,q is strictly monotone.
In fact, for anyu,v∈W1,p(Ω),
(3.5)(u-v,Bp,qu-Bp,qv)=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u-(C(x)+|∇v|2)(p-2)/2∇v,∇u-∇v〉dx+ε∫Ω‍(|u|q-2u-|v|q-2v)(u-v)dx≥∫Ω‍[(C(x)+|∇u|2)(p-2)/2|∇u|2-(C(x)+|∇u|2)(p-2)/2|∇u||∇v|-(C(x)+|∇v|2)(p-2)/2|∇v||∇u|+(C(x)+|∇v|2)(p-2)/2|∇v|2]dx+ε∫Ω‍(|u|q-|u|q-1|v|-|v|q-1|u|+|v|q)dx=∫Ω‍[(C(x)+|∇u|2)(p-2)/2|∇u|-(C(x)+|∇v|2)(p-2)/2|∇v|](|∇u|-|∇v|)dx+ε∫Ω‍(|u|q-1-|v|q-1)(|u|-|v|)dx.
Defineh(t)=(k+t2)(p-2)/2t, for all t>0, where k is a nonnegative constant.
Case
1. If p≥2, then h′(t)=(p-2)(k+t2)p/2-2t2+(k+t2)p/2-1>0, for all t>0. That is, h:R→R is strictly increasing. Thus we can see from (3.5) that if (u-v,Bp,qu-Bp,qv)=0, then u(x)=v(x) and ∂u/∂xi=∂v/∂xi(i=1,2,…,N), a.e. in Ω, which implies that u(x)=v(x) in W1,p(Ω). Therefore, Bp,q is strictly monotone.
Case
2. If 2N/(N+1)<p<2, then for all t>0,
(3.6)h′(t)=(p-2)(k+t2)p/2-2t2+(k+t2)p/2-1=(p-2)(k+t2)p/2-1-(p-2)(k+t2)p/2-2k+(k+t2)p/2-1=(p-1)(k+t2)p/2-1-(p-2)(k+t2)p/2-2k=(p-1)(k+t2)p/2-1+(2-p)(k+t2)p/2-2k>0,
which implies that h:R→R is also strictly increasing. In the same way as Case 1, we know that Bp,q is strictly monotone.
This completes the proof.Lemma 3.2 (see Wei and Agarwal [7]).
The mappingΦp:W1,p(Ω)→R defined by Φp(u)=∫Γ‍φx(u|Γ(x))dΓ(x), for any u∈W1,p(Ω), is proper, convex, and lower semicontinuous on W1,p(Ω).Therefore,∂Φp is maximal monotone in view of Lemma 2.3.Lemma 3.3.
Define the mappingF:W1,p(Ω)→(W1,p(Ω))* by
(3.7)(v,Fu(x))=∫Ω‍g(x,u(x),∇u(x))v(x)dx,
for all u,v∈W1,p(Ω); then F is everywhere defined, monotone, and hemicontinuous on W1,p(Ω).Moreover, Lemma2.2 implies that F is maximal monotone.Proof.
We split our proof into four steps.
Step
1. For u∈W1,p(Ω),  x→g(x,u(x),∇u(x)) is measurable on Ω.
From the facts thatu(x),∂u/∂xi∈Lp(Ω),  i=1,2,…,N, we know that x→(u(x),∂u/∂x1,…,∂u/∂xN) is measurable on Ω. Combining with the fact that g satisfies Carathéodory’s conditions, we know that x→g(x,u(x),∇u(x)) is measurable on Ω.
Step
2.
F is everywhere defined.
From Lemma2.1, we know that W1,p(Ω)↪CB(Ω), when p>N, and W1,p(Ω)↪Lp'(Ω), when p=N. Thus, for all w∈W1,p(Ω),
(3.8)∥w∥p′≤k∥w∥1,p,Ω,
where k>0 is a constant.
Whenp<N, we know from Lemma 2.1 that W1,p(Ω)↪LNp/(N-p)(Ω). Since 2N/(N+1)<p<+∞, then Np/(N-p)>p′ and LNp/(N-p)(Ω)↪Lp′(Ω), which implies that (3.8) is still true.
Now, foru,v∈W1,p(Ω), we have from (3.8) that
(3.9)|(v,Fu)|≤∫Ω‍|h1(x)||v|dx+b∫Ω‍|u|p-1|v|dx+b∑i=1N‍∫Ω‍|∂u∂xi|p-1|v|dx≤∥h1(x)∥p∥v∥p′+b∥u∥pp/p′∥v∥p+b∑i=1N‍∥∂u∂xi∥pp/p′∥v∥p≤∥h1(x)∥p∥v∥1,p,Ω+b(N+1)∥u∥1,p,Ωp/p′∥v∥1,p,Ω,
which implies that F is everywhere defined.
Step
3.
F is monotone.
Sinceg(x,r1,…,rN+1) is monotone with respect to r1, then F is monotone.
Step
4.
F is hemicontinuous.
In fact, it suffices to show that, for anyu,v,w∈W1,p(Ω) and t∈[0,1],  (w,F(u+tv)-Fu)→0, as t→0.
Noticing the facts thatg is measurable on Ω and g satisfies Carathéodory’s conditions, by using Lebesque’s dominated convergence theorem, we have
(3.10)0≤limt→0|(w,F(u+tv)-Fu)|≤∫Ω‍limt→0|g(x,u(x)+tv(x),∇u+t∇v)-g(x,u,∇u)||w|dx=0,
and hence F is hemicontinuous.
This completes the proof.In view of Lemma2.4, we can easily obtain the following result.Lemma 3.4.
B
p
,
q
+
F
+
∂
Φ
p
:
W
1
,
p
(
Ω
)
→
(
W
1
,
p
(
Ω
)
)
* is maximal monotone.Lemma 3.5.
Define the mappingS:Lp(Γ)→Lp'(Γ) by Su=βx(u(x)), for any u∈Lp(Γ). Define the mapping K:W1,p(Ω)→Lp(Γ) by Kv=v|Γ for any v∈W1,p(Ω). Then K*SK=∂Φp, where Φp is the same as that in Lemma 3.2.Proof.
We will prove the result under the additional condition|βx(u)|≤a|u|p/p′+b(x), where b(x)∈Lp′(Γ) and a∈R. Refer to the result of Brezis [15] for the general case.
It is obvious thatS is continuous. For all u(x),v(x)∈Lp(Γ), since βx is monotone, then (u-v,Su-Sv)=∫Γ‍(βx(u(x))-βx(v(x)))(u(x)-v(x))dΓ(x)≥0, which implies that S is monotone. Thus, S:Lp(Γ)→Lp′(Γ) is maximal monotone in view of Lemma 2.2.
DefineΨ:Lp(Γ)→R by Ψ(u)=∫Γ‍φx(u(x))dΓ(x), for all u(x)∈Lp(Γ); then it is easy to see that Ψ is a proper, convex, and lower semicontinuous function on Lp(Γ), which implies that ∂Ψ:Lp(Γ)→Lp′(Γ) is maximal monotone. Since Ψ(u)-Ψ(v)=∫Γ‍[φx(u(x))-φx(v(x))]dΓ(x)≥∫Γ‍βx(v(x))(u(x)-v(x))dΓ(x)=(Sv,u-v), for all u(x),v(x)∈Lp(Γ), then Bv∈∂Ψ(v). So S=∂Ψ.
Now clearly,K*SK:W1,p(Ω)→(W1,p(Ω))* is maximal monotone since both K,S are continuous. Finally, for any u,v∈W1,p(Ω), we have
(3.11)Φp(v)-Φp(u)=Ψ(Kv)-Ψ(Ku)=∫Γ[φx(v|Γ(x))-φx(u|Γ(x))]dΓ(x)≥∫Γβx(u|Γ(x))(v|Γ(x)-u|Γ(x))dΓ(x)=(SKu,Kv-Ku)=(K*SKu,v-u).
Hence we get K*SK⊂∂Φp and so K*SK=∂Φp.
This completes the proof.Lemma 3.6.
One hasR(Bp,q+F+∂Φp)=(W1,p(Ω))*.Proof.
From Lemma3.5, we know that for all u∈W1,p(Ω),
(3.12)(u,∂Φp(u))=∫Γ‍βx(u|Γ(x))u|Γ(x)dΓ(x).
Since 0∈βx(0), then for all u∈W1,p(Ω),  (u,∂Φp(u))≥0.
SinceF is monotone, then (u,Fu-F0)≥0. Moreover, in view of (3.8), we have
(3.13)|(u,F0)|∥u∥1,p,Ω≤∫Ω|h1(x)||u(x)|dx∥u∥1,p,Ω≤∥h1(x)∥p∥u∥p′∥u∥1,p,Ω≤k∥h1(x)∥p<+∞.
Then by using Lemma 3.1, we have
(3.14)(u,Bp,qu+Fu+∂Φp(u))∥u∥1,p,Ω≥(u,Bp,qu)∥u∥1,p,Ω+(u,F0)∥u∥1,p,Ω⟶+∞,
as ∥u∥1,p,Ω→+∞, which implies that Bp,q+F+∂Φp is coercive. Then Lemmas 3.4 and 2.5 ensure that the result is true.
This completes the proof.Theorem 3.7.
Forf∈Lp′(Ω), nonlinear boundary value problem (1.4) has a unique solution u∈W1,p(Ω).Proof.
From Lemma3.6, we know that for f∈Lp′(Ω), there exists u∈W1,p(Ω) such that
(3.15)f=Bp,qu+∂Φp(u)+Fu.
Now, we will show that thisu is unique.
Otherwise, there existsv∈W1,p(Ω) satisfying (3.15). Then notice the facts that Bp,q,∂Φp and F are all monotone, we have
(3.16)0≤(u-v,Bp,qu-Bp,qv)=(u-v,f-Fu-∂Φp(u)-f+Fv+∂Φp(v))=(u-v,Fv-Fu)+(u-v,∂Φp(v)-∂Φp(u))≤0.
Since Bp,q is strictly monotone, then u=v.
Next, we will show that thisu is the solution of (1.4).
SinceΦp(u+φ)=Φp(u) for any u∈W1,p(Ω) and φ∈C0∞(Ω), where 2N/(N+1)<p<+∞ and N≥1, we have (φ,∂Φ(u))=0. Since f-Fu=Bp,qu+∂Φp(u), then for all φ∈C0∞(Ω),
(3.17)∫Ω(f-g(x,u(x),∇u(x)))φdx=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇φ〉dx+ε∫Ω|u(x)|q-2u(x)φ(x)dx=-∫Ωdiv[(C(x)+|∇u|2)(p-2)/2∇u]φdx+ε∫Ω|u(x)|q-2u(x)φdx,
which implies that the result
(3.18)-div[(C(x)+|∇u|2)(p-2)/2∇u]+ε|u|q-2u+g(x,u(x),∇u(x))=f(x)a.e.x∈Ω
is true.
From (3.18), we know that f(x)=-div[(C(x)+|∇u|2)(p-2)/2∇u]+ε|u|q-2u+g(x,u(x),∇u(x))∈Lp′(Ω). By using Green's formula, we have that for any v∈W1,p(Ω),
(3.19)∫Γ〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉v|ΓdΓ(x)=∫Ωdiv[(C(x)+|∇u|2)(p-2)/2∇u]vdx+∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇v〉dx.
Then-〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉∈W-1/p,p′(Γ)=(W1/p,p(Γ))*, where W1/p,p(Γ) is the space of the traces of W1,p(Ω).
Combining with the results of Lemma3.5, (3.18), and (3.19), we have
(3.20)∫Γ〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉v|ΓdΓ(x)=(v,ε|u(x)|q-2u(x)+g(x,u(x),∇u(x))-f)+(v,Bpu-ε|u(x)|q-2u(x))=(v,Bpu+Fu-f)=(v,-∂Φp(u))=(v,-K*SKu)=-(Kv,SKu)=-∫Γβx(u|Γ(x))v|Γ(x)dΓ(x).
Then(3.21)-〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉∈βx(u(x)),a.e.onΓ.
From (3.19) and (3.21) we know that u is the solution of (1.4).
This completes the proof.


### 3.3. Iterative Construction of the Solution of (1.4)

Lemma 3.8 (Wei and Zhou [16]).
Suppose thatX is a real smooth and uniformly convex Banach space and A:X→2X* is a maximal monotone operator with A-1(0)≠∅. Let {xn} be a sequence generated by the following iterative scheme:
(3.22)x1∈X,r1>0,xn+1=J-1[βnJx1+(1-βn)J(J+rnA)-1Jxn],∀n≥1.
If{rn}⊂(0,+∞) with limn→∞rn=+∞,  {βn}⊂[0,1] with ∑n=1∞‍βn=+∞, and βn→0 as n→∞, then {xn} converges strongly to ΠA-1(0)(x1), where ΠA-1(0) is the generalized projection operator from X onto A-1(0).Definition 3.9.
Define the mappingCp,q:W1,p(Ω)→(W1,p(Ω))* by
(3.23)(v,Cp,qu)=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇v〉dx+ε∫Ω|u(x)|q-2u(x)v(x)dx-∫Ωf(x)v(x)dx+∫Ωg(x,u(x),∇u(x))v(x)dx+(v,∂Φp(u))
for any u,v∈W1,p(Ω), where f∈Lp'(Ω) is the same as that in (1.4).Similarly to Lemma3.4, we know that Cp,q is also a maximal monotone operator. Moreover, we can easily get the following result.Lemma 3.10.
u
∈
W
1
,
p
(
Ω
) is the solution of (1.4) if and only if u∈W1,p(Ω) is the zero point of Cp,q.Proof.
Letu(x) be the solution of (1.4), then for all v∈W1,p(Ω), by using Green’s formula and Lemma 3.5, we have
(3.24)(v,Cp,qu)=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇v〉dx+ε∫Ω|u(x)|q-2u(x)v(x)dx-∫Ωf(x)v(x)dx+∫Ωg(x,u(x),∇u(x))v(x)dx+(v,∂Φp(u))=-∫Ωdiv[(C(x)+|∇u|2)(p-2)/2∇u]vdx+∫Γ〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉v|ΓdΓ(x)+ε∫Ω|u(x)|q-2u(x)v(x)dx-∫Ωf(x)v(x)dx+∫Ωg(x,u(x),∇u(x))v(x)dx+(v,K*SKu)=∫Γ〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉v|ΓdΓ(x)+(v,K*SKu)=-∫Γβx(u|Γ)v|ΓdΓ(x)+∫Γβx(u|Γ)v|ΓdΓ(x)=0.
Thusu∈Cp,q-1(0).
Ifu∈Cp,q-1(0), then for all φ∈C0∞(Ω),
(3.25)0=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇φ〉dx+ε∫Ω|u(x)|q-2u(x)φ(x)dx-∫Ωfφdx+∫Ωg(x,u(x),∇u(x))φdx,
which implies the result-div[(C(x)+|∇u|2)(p-2)/2∇u]+ε|u(x)|q-2u(x)+g(x,u(x),∇u(x))=f(x), a.e.x∈Ω is true.
Copying the last part of Theorem3.7, we can obtain (3.21), which implies that u is the solution of (1.4).
This completes the proof.Remark 3.11.
From Lemma3.10 we can see that Cp,q-1(0)≠∅. It is a good example to show that the assumption that A-1(0)≠∅ in Lemma 3.8 is valid.Lemma 3.12 (Takahashi [17]).
LetX be a Banach space and J a duality mapping defined on X. If J is single valued, then X is smooth.Based on the facts of Lemmas3.8, 3.10, and 3.12, we can construct an iterative sequence to approximate strongly to the solution of (1.4).Theorem 3.13.
Let{un(x)} be a sequence generated by the following iterative scheme:
(3.26)u1(x)∈W1,p(Ω),r1>0,chosenarbitrarily,un+1(x)=J-1[βnJu1(x)+(1-βn)J(J+rnCp,q)-1Jun(x)],∀n≥1.
If{rn}⊂(0,+∞) with limn→∞rn=+∞,  {βn}⊂[0,1] with ∑n=1∞‍βn=+∞, and βn→0 as n→∞, then {un(x)} converges strongly to ΠCp,q-1(0)(u1(x)).Remark 3.14.
Theorem3.13 not only tells us that the sequence {un(x)} generated by (3.26) converges strongly to the solution of (1.4), but also tells us that the unique solution of (1.4) is the generalized projection of the initial function u1(x) onto Cp,q-1(0).Remark 3.15.
Compared to the work done in [8], we may find that different techniques are employed to show the existence and uniqueness of the solution of the desired equation because the nonlinear item g(x,u(x),∇u(x)) is involved.Remark 3.16.
We can get the following special cases in our paper.Corollary 3.17.
Forf∈L2(Ω), the following equation has a unique solution in H1(Ω):
(3.27)-Δu+ε|u|q-2u+g(x,u(x),∇u(x))=f(x),a.e.inΩ,-∂u∂ϑ∈βx(u(x)),a.e.onΓ.Corollary 3.18.
Let{un(x)} be a sequence generated by
(3.28)u1(x)∈H1(Ω),r1>0,chosenarbitrarily,un+1(x)=βnu1(x)+(1-βn)(I+rnC2,q)-1un(x),∀n≥1,
where C2,q is a special case of Cp,q defined in Definition 3.9 if p≡2.
If{rn}⊂(0,+∞) with limn→∞rn=+∞,  {βn}⊂[0,1] with ∑n=1∞‍βn=+∞, and βn→0 as n→∞, then {un(x)} converges strongly to PC2,q-1(0)(u1(x)), where PC2,q-1(0)(u1(x)) denotes the metric projection from H1(Ω) onto C2,q-1(0). And,  PC2,q-1(0)(u1(x)) is the unique solution of (3.27).Corollary 3.19.
Forf∈Lp′(Ω), the following equation has a unique solution in W1,p(Ω):(3.29)-Δpu+ε|u|q-2u+g(x,u(x),∇u(x))=f(x),a.e.inΩ,-〈ϑ,|∇u|p-2∇u〉∈βx(u(x)),a.e.onΓ.Corollary 3.20.
Let{un(x)} be a sequence generated by (3.26), where Cp,q:W1,p(Ω)→(W1,p(Ω))* is defined by(3.30)(v,Cp,qu)=∫Ω‍〈|∇u|p-2∇u,∇v〉dx+ε∫Ω‍|u(x)|q-2u(x)v(x)dx-∫Ω‍f(x)v(x)dx+∫Ω‍g(x,u(x),∇u(x))v(x)dx+(v,∂Φp(u)),
for u(x),v(x)∈W1,p(Ω).Then under the conditions of Theorem3.13, we know that {un(x)} converges strongly to both the unique solution of (3.29) and the zero point of Cp,q.


## 3.1. Notations and Assumptions of (1.4)

In the following of this paper, unless otherwise stated, we will assume that2N/(N+1)<p<+∞,  1≤q<+∞ if p≥N, and 1≤q≤Np/(N-p) if p<N, for N≥1. Let 1/p+1/p′=1. We use ∥·∥p,∥·∥p', and ∥·∥1,p,Ω to denote the norm of spaces Lp(Ω),Lp′(Ω), and W1,p(Ω), respectively.In nonlinear boundary value problem (1.4), Ω is a bounded conical domain of an Euclidean space RN with its boundary Γ∈C1 (see Wei and He [1]). We will assume that Green’s formula is available. f(x)∈Lp′(Ω) is a given function. 0≤C(x)∈Lp(Ω),  ε is a nonnegative constant and ϑ denotes the exterior normal derivative of Γ.Letφ:Γ×R→R be a given function such that, for each x∈Γ,  φx=φ(x,·):R→R is a proper, convex, and lower semicontinuous function with φx(0)=0. Let βx be the subdifferential of φx, that is, βx≡∂φx. Suppose that 0∈βx(0) and for each t∈R, the function x∈Γ→(I+λβx)-1(t)∈R is measurable for λ>0.Suppose thatg:Ω×RN+1→R is a given function satisfying the following conditions, which can be found in Zeidler [14].(a)
Carathéodory's conditions:(3.1)x⟶g(x,r)ismeasurableonΩ∀r∈RN+1,r⟶g(x,r)iscontinuousonRN+1foralmostallx∈Ω.(b)
Growth condition:(3.2)g(x,r1,…,rN+1)≤h1(x)+b∑i=1N+1|ri|p-1,
where (r1,r2,…,rN+1)∈RN+1,  h1(x)∈Lp(Ω) and b is a fixed positive constant.(c)
Monotone condition:g is monotone with respect to r1, that is,
(3.3)(g(x,r1,…,rN+1)-g(x,t1,…,tN+1))(r1-t1)≥0,
for all x∈Ω and (r1,…,rN+1),(t1,…,tN+1)∈RN+1.


## 3.2. Existence and Uniqueness of the Solution of (1.4)

Lemma 3.1.
Define the mappingBp,q:W1,p(Ω)→(W1,p(Ω))* by
(3.4)(v,Bp,qu)=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇v〉dx+ε∫Ω‍|u(x)|q-2u(x)v(x)dx
for any u,v∈W1,p(Ω). Then, Bp,q is everywhere defined, strictly monotone, hemicontinuous, and coercive.Moreover, Lemma2.2 implies that Bp,q is maximal monotone.Proof.
From Lemma  3.2 by Wei and Agarwal [7], we know that Bp,q is everywhere defined, monotone, hemicontinuous, and coercive. Then we only need to show that Bp,q is strictly monotone.
In fact, for anyu,v∈W1,p(Ω),
(3.5)(u-v,Bp,qu-Bp,qv)=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u-(C(x)+|∇v|2)(p-2)/2∇v,∇u-∇v〉dx+ε∫Ω‍(|u|q-2u-|v|q-2v)(u-v)dx≥∫Ω‍[(C(x)+|∇u|2)(p-2)/2|∇u|2-(C(x)+|∇u|2)(p-2)/2|∇u||∇v|-(C(x)+|∇v|2)(p-2)/2|∇v||∇u|+(C(x)+|∇v|2)(p-2)/2|∇v|2]dx+ε∫Ω‍(|u|q-|u|q-1|v|-|v|q-1|u|+|v|q)dx=∫Ω‍[(C(x)+|∇u|2)(p-2)/2|∇u|-(C(x)+|∇v|2)(p-2)/2|∇v|](|∇u|-|∇v|)dx+ε∫Ω‍(|u|q-1-|v|q-1)(|u|-|v|)dx.
Defineh(t)=(k+t2)(p-2)/2t, for all t>0, where k is a nonnegative constant.
Case
1. If p≥2, then h′(t)=(p-2)(k+t2)p/2-2t2+(k+t2)p/2-1>0, for all t>0. That is, h:R→R is strictly increasing. Thus we can see from (3.5) that if (u-v,Bp,qu-Bp,qv)=0, then u(x)=v(x) and ∂u/∂xi=∂v/∂xi(i=1,2,…,N), a.e. in Ω, which implies that u(x)=v(x) in W1,p(Ω). Therefore, Bp,q is strictly monotone.
Case
2. If 2N/(N+1)<p<2, then for all t>0,
(3.6)h′(t)=(p-2)(k+t2)p/2-2t2+(k+t2)p/2-1=(p-2)(k+t2)p/2-1-(p-2)(k+t2)p/2-2k+(k+t2)p/2-1=(p-1)(k+t2)p/2-1-(p-2)(k+t2)p/2-2k=(p-1)(k+t2)p/2-1+(2-p)(k+t2)p/2-2k>0,
which implies that h:R→R is also strictly increasing. In the same way as Case 1, we know that Bp,q is strictly monotone.
This completes the proof.Lemma 3.2 (see Wei and Agarwal [7]).
The mappingΦp:W1,p(Ω)→R defined by Φp(u)=∫Γ‍φx(u|Γ(x))dΓ(x), for any u∈W1,p(Ω), is proper, convex, and lower semicontinuous on W1,p(Ω).Therefore,∂Φp is maximal monotone in view of Lemma 2.3.Lemma 3.3.
Define the mappingF:W1,p(Ω)→(W1,p(Ω))* by
(3.7)(v,Fu(x))=∫Ω‍g(x,u(x),∇u(x))v(x)dx,
for all u,v∈W1,p(Ω); then F is everywhere defined, monotone, and hemicontinuous on W1,p(Ω).Moreover, Lemma2.2 implies that F is maximal monotone.Proof.
We split our proof into four steps.
Step
1. For u∈W1,p(Ω),  x→g(x,u(x),∇u(x)) is measurable on Ω.
From the facts thatu(x),∂u/∂xi∈Lp(Ω),  i=1,2,…,N, we know that x→(u(x),∂u/∂x1,…,∂u/∂xN) is measurable on Ω. Combining with the fact that g satisfies Carathéodory’s conditions, we know that x→g(x,u(x),∇u(x)) is measurable on Ω.
Step
2.
F is everywhere defined.
From Lemma2.1, we know that W1,p(Ω)↪CB(Ω), when p>N, and W1,p(Ω)↪Lp'(Ω), when p=N. Thus, for all w∈W1,p(Ω),
(3.8)∥w∥p′≤k∥w∥1,p,Ω,
where k>0 is a constant.
Whenp<N, we know from Lemma 2.1 that W1,p(Ω)↪LNp/(N-p)(Ω). Since 2N/(N+1)<p<+∞, then Np/(N-p)>p′ and LNp/(N-p)(Ω)↪Lp′(Ω), which implies that (3.8) is still true.
Now, foru,v∈W1,p(Ω), we have from (3.8) that
(3.9)|(v,Fu)|≤∫Ω‍|h1(x)||v|dx+b∫Ω‍|u|p-1|v|dx+b∑i=1N‍∫Ω‍|∂u∂xi|p-1|v|dx≤∥h1(x)∥p∥v∥p′+b∥u∥pp/p′∥v∥p+b∑i=1N‍∥∂u∂xi∥pp/p′∥v∥p≤∥h1(x)∥p∥v∥1,p,Ω+b(N+1)∥u∥1,p,Ωp/p′∥v∥1,p,Ω,
which implies that F is everywhere defined.
Step
3.
F is monotone.
Sinceg(x,r1,…,rN+1) is monotone with respect to r1, then F is monotone.
Step
4.
F is hemicontinuous.
In fact, it suffices to show that, for anyu,v,w∈W1,p(Ω) and t∈[0,1],  (w,F(u+tv)-Fu)→0, as t→0.
Noticing the facts thatg is measurable on Ω and g satisfies Carathéodory’s conditions, by using Lebesque’s dominated convergence theorem, we have
(3.10)0≤limt→0|(w,F(u+tv)-Fu)|≤∫Ω‍limt→0|g(x,u(x)+tv(x),∇u+t∇v)-g(x,u,∇u)||w|dx=0,
and hence F is hemicontinuous.
This completes the proof.In view of Lemma2.4, we can easily obtain the following result.Lemma 3.4.
B
p
,
q
+
F
+
∂
Φ
p
:
W
1
,
p
(
Ω
)
→
(
W
1
,
p
(
Ω
)
)
* is maximal monotone.Lemma 3.5.
Define the mappingS:Lp(Γ)→Lp'(Γ) by Su=βx(u(x)), for any u∈Lp(Γ). Define the mapping K:W1,p(Ω)→Lp(Γ) by Kv=v|Γ for any v∈W1,p(Ω). Then K*SK=∂Φp, where Φp is the same as that in Lemma 3.2.Proof.
We will prove the result under the additional condition|βx(u)|≤a|u|p/p′+b(x), where b(x)∈Lp′(Γ) and a∈R. Refer to the result of Brezis [15] for the general case.
It is obvious thatS is continuous. For all u(x),v(x)∈Lp(Γ), since βx is monotone, then (u-v,Su-Sv)=∫Γ‍(βx(u(x))-βx(v(x)))(u(x)-v(x))dΓ(x)≥0, which implies that S is monotone. Thus, S:Lp(Γ)→Lp′(Γ) is maximal monotone in view of Lemma 2.2.
DefineΨ:Lp(Γ)→R by Ψ(u)=∫Γ‍φx(u(x))dΓ(x), for all u(x)∈Lp(Γ); then it is easy to see that Ψ is a proper, convex, and lower semicontinuous function on Lp(Γ), which implies that ∂Ψ:Lp(Γ)→Lp′(Γ) is maximal monotone. Since Ψ(u)-Ψ(v)=∫Γ‍[φx(u(x))-φx(v(x))]dΓ(x)≥∫Γ‍βx(v(x))(u(x)-v(x))dΓ(x)=(Sv,u-v), for all u(x),v(x)∈Lp(Γ), then Bv∈∂Ψ(v). So S=∂Ψ.
Now clearly,K*SK:W1,p(Ω)→(W1,p(Ω))* is maximal monotone since both K,S are continuous. Finally, for any u,v∈W1,p(Ω), we have
(3.11)Φp(v)-Φp(u)=Ψ(Kv)-Ψ(Ku)=∫Γ[φx(v|Γ(x))-φx(u|Γ(x))]dΓ(x)≥∫Γβx(u|Γ(x))(v|Γ(x)-u|Γ(x))dΓ(x)=(SKu,Kv-Ku)=(K*SKu,v-u).
Hence we get K*SK⊂∂Φp and so K*SK=∂Φp.
This completes the proof.Lemma 3.6.
One hasR(Bp,q+F+∂Φp)=(W1,p(Ω))*.Proof.
From Lemma3.5, we know that for all u∈W1,p(Ω),
(3.12)(u,∂Φp(u))=∫Γ‍βx(u|Γ(x))u|Γ(x)dΓ(x).
Since 0∈βx(0), then for all u∈W1,p(Ω),  (u,∂Φp(u))≥0.
SinceF is monotone, then (u,Fu-F0)≥0. Moreover, in view of (3.8), we have
(3.13)|(u,F0)|∥u∥1,p,Ω≤∫Ω|h1(x)||u(x)|dx∥u∥1,p,Ω≤∥h1(x)∥p∥u∥p′∥u∥1,p,Ω≤k∥h1(x)∥p<+∞.
Then by using Lemma 3.1, we have
(3.14)(u,Bp,qu+Fu+∂Φp(u))∥u∥1,p,Ω≥(u,Bp,qu)∥u∥1,p,Ω+(u,F0)∥u∥1,p,Ω⟶+∞,
as ∥u∥1,p,Ω→+∞, which implies that Bp,q+F+∂Φp is coercive. Then Lemmas 3.4 and 2.5 ensure that the result is true.
This completes the proof.Theorem 3.7.
Forf∈Lp′(Ω), nonlinear boundary value problem (1.4) has a unique solution u∈W1,p(Ω).Proof.
From Lemma3.6, we know that for f∈Lp′(Ω), there exists u∈W1,p(Ω) such that
(3.15)f=Bp,qu+∂Φp(u)+Fu.
Now, we will show that thisu is unique.
Otherwise, there existsv∈W1,p(Ω) satisfying (3.15). Then notice the facts that Bp,q,∂Φp and F are all monotone, we have
(3.16)0≤(u-v,Bp,qu-Bp,qv)=(u-v,f-Fu-∂Φp(u)-f+Fv+∂Φp(v))=(u-v,Fv-Fu)+(u-v,∂Φp(v)-∂Φp(u))≤0.
Since Bp,q is strictly monotone, then u=v.
Next, we will show that thisu is the solution of (1.4).
SinceΦp(u+φ)=Φp(u) for any u∈W1,p(Ω) and φ∈C0∞(Ω), where 2N/(N+1)<p<+∞ and N≥1, we have (φ,∂Φ(u))=0. Since f-Fu=Bp,qu+∂Φp(u), then for all φ∈C0∞(Ω),
(3.17)∫Ω(f-g(x,u(x),∇u(x)))φdx=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇φ〉dx+ε∫Ω|u(x)|q-2u(x)φ(x)dx=-∫Ωdiv[(C(x)+|∇u|2)(p-2)/2∇u]φdx+ε∫Ω|u(x)|q-2u(x)φdx,
which implies that the result
(3.18)-div[(C(x)+|∇u|2)(p-2)/2∇u]+ε|u|q-2u+g(x,u(x),∇u(x))=f(x)a.e.x∈Ω
is true.
From (3.18), we know that f(x)=-div[(C(x)+|∇u|2)(p-2)/2∇u]+ε|u|q-2u+g(x,u(x),∇u(x))∈Lp′(Ω). By using Green's formula, we have that for any v∈W1,p(Ω),
(3.19)∫Γ〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉v|ΓdΓ(x)=∫Ωdiv[(C(x)+|∇u|2)(p-2)/2∇u]vdx+∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇v〉dx.
Then-〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉∈W-1/p,p′(Γ)=(W1/p,p(Γ))*, where W1/p,p(Γ) is the space of the traces of W1,p(Ω).
Combining with the results of Lemma3.5, (3.18), and (3.19), we have
(3.20)∫Γ〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉v|ΓdΓ(x)=(v,ε|u(x)|q-2u(x)+g(x,u(x),∇u(x))-f)+(v,Bpu-ε|u(x)|q-2u(x))=(v,Bpu+Fu-f)=(v,-∂Φp(u))=(v,-K*SKu)=-(Kv,SKu)=-∫Γβx(u|Γ(x))v|Γ(x)dΓ(x).
Then(3.21)-〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉∈βx(u(x)),a.e.onΓ.
From (3.19) and (3.21) we know that u is the solution of (1.4).
This completes the proof.


## 3.3. Iterative Construction of the Solution of (1.4)

Lemma 3.8 (Wei and Zhou [16]).
Suppose thatX is a real smooth and uniformly convex Banach space and A:X→2X* is a maximal monotone operator with A-1(0)≠∅. Let {xn} be a sequence generated by the following iterative scheme:
(3.22)x1∈X,r1>0,xn+1=J-1[βnJx1+(1-βn)J(J+rnA)-1Jxn],∀n≥1.
If{rn}⊂(0,+∞) with limn→∞rn=+∞,  {βn}⊂[0,1] with ∑n=1∞‍βn=+∞, and βn→0 as n→∞, then {xn} converges strongly to ΠA-1(0)(x1), where ΠA-1(0) is the generalized projection operator from X onto A-1(0).Definition 3.9.
Define the mappingCp,q:W1,p(Ω)→(W1,p(Ω))* by
(3.23)(v,Cp,qu)=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇v〉dx+ε∫Ω|u(x)|q-2u(x)v(x)dx-∫Ωf(x)v(x)dx+∫Ωg(x,u(x),∇u(x))v(x)dx+(v,∂Φp(u))
for any u,v∈W1,p(Ω), where f∈Lp'(Ω) is the same as that in (1.4).Similarly to Lemma3.4, we know that Cp,q is also a maximal monotone operator. Moreover, we can easily get the following result.Lemma 3.10.
u
∈
W
1
,
p
(
Ω
) is the solution of (1.4) if and only if u∈W1,p(Ω) is the zero point of Cp,q.Proof.
Letu(x) be the solution of (1.4), then for all v∈W1,p(Ω), by using Green’s formula and Lemma 3.5, we have
(3.24)(v,Cp,qu)=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇v〉dx+ε∫Ω|u(x)|q-2u(x)v(x)dx-∫Ωf(x)v(x)dx+∫Ωg(x,u(x),∇u(x))v(x)dx+(v,∂Φp(u))=-∫Ωdiv[(C(x)+|∇u|2)(p-2)/2∇u]vdx+∫Γ〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉v|ΓdΓ(x)+ε∫Ω|u(x)|q-2u(x)v(x)dx-∫Ωf(x)v(x)dx+∫Ωg(x,u(x),∇u(x))v(x)dx+(v,K*SKu)=∫Γ〈ϑ,(C(x)+|∇u|2)(p-2)/2∇u〉v|ΓdΓ(x)+(v,K*SKu)=-∫Γβx(u|Γ)v|ΓdΓ(x)+∫Γβx(u|Γ)v|ΓdΓ(x)=0.
Thusu∈Cp,q-1(0).
Ifu∈Cp,q-1(0), then for all φ∈C0∞(Ω),
(3.25)0=∫Ω〈(C(x)+|∇u|2)(p-2)/2∇u,∇φ〉dx+ε∫Ω|u(x)|q-2u(x)φ(x)dx-∫Ωfφdx+∫Ωg(x,u(x),∇u(x))φdx,
which implies the result-div[(C(x)+|∇u|2)(p-2)/2∇u]+ε|u(x)|q-2u(x)+g(x,u(x),∇u(x))=f(x), a.e.x∈Ω is true.
Copying the last part of Theorem3.7, we can obtain (3.21), which implies that u is the solution of (1.4).
This completes the proof.Remark 3.11.
From Lemma3.10 we can see that Cp,q-1(0)≠∅. It is a good example to show that the assumption that A-1(0)≠∅ in Lemma 3.8 is valid.Lemma 3.12 (Takahashi [17]).
LetX be a Banach space and J a duality mapping defined on X. If J is single valued, then X is smooth.Based on the facts of Lemmas3.8, 3.10, and 3.12, we can construct an iterative sequence to approximate strongly to the solution of (1.4).Theorem 3.13.
Let{un(x)} be a sequence generated by the following iterative scheme:
(3.26)u1(x)∈W1,p(Ω),r1>0,chosenarbitrarily,un+1(x)=J-1[βnJu1(x)+(1-βn)J(J+rnCp,q)-1Jun(x)],∀n≥1.
If{rn}⊂(0,+∞) with limn→∞rn=+∞,  {βn}⊂[0,1] with ∑n=1∞‍βn=+∞, and βn→0 as n→∞, then {un(x)} converges strongly to ΠCp,q-1(0)(u1(x)).Remark 3.14.
Theorem3.13 not only tells us that the sequence {un(x)} generated by (3.26) converges strongly to the solution of (1.4), but also tells us that the unique solution of (1.4) is the generalized projection of the initial function u1(x) onto Cp,q-1(0).Remark 3.15.
Compared to the work done in [8], we may find that different techniques are employed to show the existence and uniqueness of the solution of the desired equation because the nonlinear item g(x,u(x),∇u(x)) is involved.Remark 3.16.
We can get the following special cases in our paper.Corollary 3.17.
Forf∈L2(Ω), the following equation has a unique solution in H1(Ω):
(3.27)-Δu+ε|u|q-2u+g(x,u(x),∇u(x))=f(x),a.e.inΩ,-∂u∂ϑ∈βx(u(x)),a.e.onΓ.Corollary 3.18.
Let{un(x)} be a sequence generated by
(3.28)u1(x)∈H1(Ω),r1>0,chosenarbitrarily,un+1(x)=βnu1(x)+(1-βn)(I+rnC2,q)-1un(x),∀n≥1,
where C2,q is a special case of Cp,q defined in Definition 3.9 if p≡2.
If{rn}⊂(0,+∞) with limn→∞rn=+∞,  {βn}⊂[0,1] with ∑n=1∞‍βn=+∞, and βn→0 as n→∞, then {un(x)} converges strongly to PC2,q-1(0)(u1(x)), where PC2,q-1(0)(u1(x)) denotes the metric projection from H1(Ω) onto C2,q-1(0). And,  PC2,q-1(0)(u1(x)) is the unique solution of (3.27).Corollary 3.19.
Forf∈Lp′(Ω), the following equation has a unique solution in W1,p(Ω):(3.29)-Δpu+ε|u|q-2u+g(x,u(x),∇u(x))=f(x),a.e.inΩ,-〈ϑ,|∇u|p-2∇u〉∈βx(u(x)),a.e.onΓ.Corollary 3.20.
Let{un(x)} be a sequence generated by (3.26), where Cp,q:W1,p(Ω)→(W1,p(Ω))* is defined by(3.30)(v,Cp,qu)=∫Ω‍〈|∇u|p-2∇u,∇v〉dx+ε∫Ω‍|u(x)|q-2u(x)v(x)dx-∫Ω‍f(x)v(x)dx+∫Ω‍g(x,u(x),∇u(x))v(x)dx+(v,∂Φp(u)),
for u(x),v(x)∈W1,p(Ω).Then under the conditions of Theorem3.13, we know that {un(x)} converges strongly to both the unique solution of (3.29) and the zero point of Cp,q.



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*Source: 210325-2012-11-19.xml*