# On thep-Biharmonic Operator with Critical Sobolev Exponent and Nonlinear Steklov Boundary Condition

Authors: Abdelouahed El Khalil; My Driss Morchid Alaoui; Abdelfattah Touzani
Journal: International Journal of Analysis
 (2014)

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## Abstract

We show that this operator possesses at least one nondecreasing sequence of positive eigenvalues. A direct characterization of the principal eigenvalue (the first one) is given that we apply to study the spectrum of thep-biharmonic operator with a critical Sobolev exponent and the nonlinear Steklov boundary conditions using variational arguments and trace critical Sobolev embedding.


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## Body

## 1. Introduction

LetΩ be a smooth bounded domain in ℝN. Consider the fourth-order nonlinear Steklov boundary eigenvalue problem
(Ep)λΔp2u=|u|p*-2uinΩ,|Δu|p-2Δu=λ|∂u∂n|p-2∂u∂non⁡Γmmmmlllu∈W2,p(Ω)∩W01,p(Ω),
where p* is the critical Sobolev exponent defined by
(1)p*=NpN-2pwith2<2p<N.
Here λ is a real parameter which plays the role of an eigenvalue.Δ
p
2
u
:
=
Δ
(
|
Δ
u
|
p
-
2
Δ
u
) is the operator of fourth order called the p-biharmonic operator. For p=2, the linear operator Δ22=Δ2=Δ. Δ is the iterated Laplacian that multiplied with positive constant appears often in Navier-Stokes equations as being a viscosity coefficient. Its reciprocal operator, denoted by (Δ2)-1, is celebrated Green’s operator [1].The nonlinear boundary condition describes a nonlinear flux through the boundaryΓ which depends on the solution itself and its normal derivation. Here ∂u/∂n denotes the outer normal derivative of u on Γ defined by ∂u/∂n=∇u·n→.Notice that the biharmonic equationΔ2u=0, corresponding to p=2, is a partial differential equation of fourth order which appears in quantum mechanics and in the theory of linear elasticity modeling Stokes’ flows. It is well known that elliptic problems with eigenvalues in the boundary conditions are usually called Steklov problems from their first appearance in [2]. For the fourth-order Steklov eigenvalue problems, the first eigenvalue plays a crucial role in the positivity preserving property for the biharmonic operator under conditions u=0, Δu-λ(∂u/∂n)=0 on Γ (see [3]). In [4], the authors investigated the bound for the first eigenvalue on the plane square and proved that the first eigenvalue is simple and its eigenfunction does not change sign. The authors of [5, 6] studied the spectrum of a fourth-order Steklov eigenvalue problem on a bounded domain in ℝN and gave the explicit form of the spectrum in the case where the domain is a ball. Let us mention that the spectrum of the fourth order Steklove has been completely determined by Ren and Yang [7] in the case p=2, using the theory of completely continuous operators.It is already evident from the well-studied second-order case that nonlinear equations with critical growth terms present highly interesting phenomena concerning the existence and nonexistence. For the fourth-order equations is more challenging, since the techniques depend strongly on the imposed boundary conditions.For the case ofp=2, p*=2N/(N-4), with N≥5, the problem (Ep=2)λ becomes linear on the right side and it is studied by Berchio et al. in [8].It is well known that fourth-order elliptic problems arise in many applications, such as microelectromechanical system, in thin film theory, nonlinear surface diffusion on solids, interface dynamics, flow in Hele-Shaw cells, phase field models of multi-phase systems, and the deformation of a nonlinear elastic beam; see, for example, [9, 10], for more details.In the nonlinear cases ofp≠2, problems governed by p-biharmonic operators attracted growing interest, and figure in variety of applications, where this operator is used to control the nonlinearity artificial viscosity of diffusion surface of non-Newtonian fluids [11].Recently, El Khalil et al. in [12] proved that the following nonlinear boundary problem:
(2)∂∂n(i|Δu|pΔu)Δp2u=|u|p-2uinΩ-∂∂n(|Δu|p-2Δu)=λ|u|p-2uon⁡Γxxxxxxxxxu∈W2,p(Ω)∩W01,p(Ω),
has at least one nondecreasing sequence of positive eigenvalues (λk)k≥1.In this paper, we use a variational technique to prove the existence of a sequence of positive eigenvalues of problem(Ep)λ.To present our result concerning(Ep)λ, we consider the homogenous problem
(Ep)λ′|Δ|ΔpuΔp2u=0inΩ|Δu|p-2Δu=λ|∂u∂n|p-2∂u∂non⁡ΓDDDDDu∈W2,p(Ω)∩W01,p(Ω).Throughout this paper, we use the notation𝕎p(Ω):=W2,p(Ω)∩W01,p(Ω).


## 2. Definitions and Preliminaries

Definition 1.
One says that a functionu∈𝕎p(Ω) is weak solution of (Ep)λ′ if
(3)∫Ω‍|Δu|p-2ΔuΔvdx=λ∫Γ‍|∂u∂n|p-2∂u∂n∂v∂nds∀v∈𝕎p(Ω).
If u is not identically zero, then one says that λ is an eigenvalue of (Ep)λ′ corresponding to the eigenfunction u.
The main objective of this work is to show that problem(Ep)λ′ has at least one nondecreasing sequence positive eigenvalues (λk)k≥1, by using a variational technique based on mini-max theory on C1-manifolds [13]. In fact, we give a direct characterization of λk involving a mini-max argument over sets of genus greater than k.We set(4)λ1=inf⁡{∥Δv∥pp,v∈𝕎p(Ω);∫Γ‍|∂u∂n|pds=1},
where ∥Δv∥p=(∫Ω‍|Δv|p)1/p denoted by ∥Δ·∥p is a norm on 𝕎p(Ω). Let us notice that 𝕎p(Ω) equipped with this norm is a uniformly convex Banach space for 1<p<+∞. The norm ∥Δ·∥p is uniformly equivalent on W02,p(Ω) to the usual norm of W02,p(Ω). Indeed, in [14] the scalar p-polyharmonic operators ΔpL, which coincide to the p-biharmonic for L=2, were recently introduced for all orders L and independently in [15] only for L even. The norms are proved to be equivalent; see also the vectorial case treated in [16]. The proof of the equivalence comes from the Poincar and Calderón-Zygmund type inequalities.For reader’s convenience, we give below the proof of the equivalence between the standard Sobolev space norm and the norm∥Δ·∥p. For that, consider the classical Dirichlet problem for the famous Poisson’s equation (see [17]):
(5)-Δu=finΩ,u=0on⁡Γ.
Let us denote by ∥u∥p=(∫Ω‍|u|pdx)1/p the norm in Lp(Ω) and by ∥u∥2,p=(∥Δu∥pp+∥u∥pp)1/p the norm in 𝕎p(Ω).IfT is the inverse operator of -Δ:𝕎p(Ω)→Lp(Ω), it is well known that (5) is uniquely solvable in 𝕎p(Ω) for all f∈Lp(Ω) and for any p∈]1,+∞[. Moreover, the operator solution
(6)T:Lp(Ω)⟶𝕎p(Ω)
satisfies the elliptic regularity estimation (continuity)
(7)∥Tf∥2,p≤Cp∥f∥p.
This allows us to say the following.Iff∈Lp(Ω) and u∈W01,p(Ω) is a solution of (5), then -Δu∈Lp(Ω), u=0 on Γ and u∈𝕎p(Ω). Moreover,
(8)∥T(-Δu)∥2,p≤Cp∥Δu∥p.
That is,
(9)∥u∥2,p≤Cp∥Δu∥p,
for a suitable positive constant Cp independent of f.On the other hand, it is easy to remark that(10)∥Δu∥p≤∥u∥2,p.
By the Closed Graph Theorem, we conclude that u→∥Δu∥p=(∫Ω‍|Δu|pdx)1/p is equivalent to the norm induced by W2,p(Ω).We see that the value defined in (4) can be written as
(11)λ1=inf⁡u∈𝕎p(Ω),u≢0∫Ω‍|Δu|pdx∫Γ‍|∂u/∂n|pds.Finally, let us point out that the problem(Ep)λ′ is naturally well defined taking in account the trace embedding 𝕎p(Ω)↪Lp(∂Ω) introduced in [12].Definition 2.
LetX be a real reflexive Banach space and let X* stand for its dual with respect to the pairing 〈·,·〉. We shall deal with mappings T acting from X into X*. The strong convergence in X (and in X*) is denoted by → and the weak convergence by ⇀. T is said to belong to the class (S+) if for any sequence un in X converging weakly to u∈X and limsup⁡n→+∞〈Δp2un,un-u〉≤0 it follows that un converges strongly to u in X. We write T∈(S+).
Consider now the following two functionals defined on𝕎p(Ω):
(12)Φ(u)=1p∫Ω‍|Δu|pdx,φ(u)=1p∫Γ‍|∂u∂n|pds,
and set 𝔐={u∈𝕎p(Ω);pφ(u)=1}.Lemma 3.
One has the following statements.(i)
Φ and φ are even and of class C1 on 𝕎p(Ω).(ii)
𝔐 is a closed C1-manifold.Proof.
It is clear thatφ and Φ are even and of class C1 on 𝕎p(Ω) and 𝔐=φ-1{1/p}. Therefore 𝔐 is closed. The derivative operator φ′ satisfies φ′(u)≠0 for all u∈𝔐 (i.e., φ′(u) is onto for all u∈𝔐), so φ is a submersion; then 𝔐 is a C1-manifold.Remark 4.
Observe thatJ:𝕎p(Ω)→(𝕎p(Ω))′ defined by
(13)J(u)={∥Δu∥p2-pΔp2uifu≠00ifu=0
is the duality mapping of 𝕎p(Ω) associated with the norm ∥Δ·∥p.The following lemma is the key of our result related to the existence.Lemma 5.
One has the following statements.(i)
φ′ is completely continuous.(ii)
The functionalΦ satisfies the Palais-Smale condition on 𝔐, that is, for {un}⊂𝔐 if {Φ(un)}n is bounded and
(14)ϵn:=Φ′(un)-gnφ′(un)⟶0asn⟶+∞,
where gn=〈Φ′(un),un〉/〈φ′(un),un〉. Then {un}n≥1 has a convergent subsequence in 𝕎p(Ω).Proof.
Letu,v∈𝕎p(Ω). We have
(15)〈φ′(u),v〉=∫Γ‍|∂u∂n|p-2∂u∂n∂v∂nds.
Then
(16)|〈φ′(u),v〉|≤∫Γ‍|∂u∂n|p-1∂v∂nds.
By applying Hölder’s inequality, we obtain
(17)|〈φ′(u),v〉|≤∥∂u∂n∥Lp′(Γ)p-1∥∂v∂n∥Lp(Ω),
where p and p′ are conjugate by the equality pp′=p+p′. Therefore
(18)|〈φ′(u),v〉|≤K∥Δu∥p′p-1∥Δv∥p.
Hence
(19)∥φ′(u)∥*≤K∥Δu∥p′p-1,
where K is the constant given by the embedding of 𝕎p(Ω) in Lp(Γ). Here ∥·∥* is the dual norm associated with ∥Δ·∥p.
Now, by the definition ofΦ we have that ∥Δ(un)∥p is bounded in ℝ.
Thus, without loss of generality, we can assume thatun converges weakly in 𝕎p(Ω) for some function u∈𝕎p(Ω) and ∥Δ(un)∥p→δ. For the rest we distinguish two cases.Ifδ=0, then un converges strongly to 0 in 𝕎p(Ω).Ifδ≠0, then let us prove that
(20)limsup⁡n→∞〈Δp2un,un-u〉≤0.
Indeed, notice that
(21)〈Δp2un,un-u〉=∥Δun∥pp-〈Δp2un,u〉.
Applying ϵn of (14) to u, we deduce that
(22)θn:=〈Δp2un,u〉-gn〈φ′(un),u〉⟶0,asn⟶∞.
Thus
(23)θn:=〈Δp2un,u〉-ϵn·〈φ′(un),u〉⟶0asn⟶∞.
Therefore
(24)〈Δp2un,un-u〉=∥Δun∥p-θn-(〈Φ′(un),un〉〈φ′(un),un〉)·〈φ′(un),u〉.
That is,
(25)〈Δp2un,un-u〉=∥Δun∥pp〈φ′(un),un〉(〈φ′(un),un〉-〈φ′(un),u〉)-θn.
On the other hand, from Lemma 5,   φ′ is completely continuous. Thus
(26)φ′(un)⟶φ′(u),〈φ′(un),un〉⟶〈φ′(u),u〉.
Then
(27)|〈φ′(un),un〉-〈φ′(un),u〉|≤|〈φ′(un),un〉-〈φ′(u),u〉|mmm+|〈φ′(un),u〉-〈φ′(u),u〉|.
It follows that
(28)|〈φ′(un),un〉-〈φ′(un),u〉|≤|〈φ′(un),un〉-〈φ′(u),u〉|mmm+∥φ′(un)-φ′(u)∥*∥Δu∥p.
This implies that
(29)〈φ′(un),un〉-〈φ′(un),u〉⟶0,asn⟶∞.
Combining with the above equalities, we obtain
(30)limsup⁡n→+∞〈Δp2un,un-u〉≤δp〈φ′(u),u〉limsup⁡n→∞(〈φ′(un),un〉-〈φ′(un),u〉).
We deduce
(31)limsup⁡n→∞〈Δp2un,un-u〉≤0.
We can write Δp2un=∥Δun∥pp-2J(un), since ∥Δun∥p≠0 for n large enough. Therefore
(32)limsup⁡n→∞〈Δp2un,un-u〉=δp-2limsup⁡n→∞〈Jun,un-u〉.
According to (31), we conclude that
(33)limsup⁡n→∞〈Jun,un-u〉≤0,
where J is the duality mapping defined in Remark 4. Thus it satisfies the condition S+ given in [18]. Therefore, un→u strongly in 𝕎p(Ω).
This achieves the proof of the lemma.


## 3. Main Results

Set(34)Γk={K⊂𝔐:Ksymmetric, compact  andγ(K)≥k},
where γ(K)=k is the genus of k, that is, the smallest integer k such that there exists an odd continuous map from K to ℝk∖{0}.Let us now state our first main result of this paper using mini-max theory; we have our main result formulated as follows.Theorem 6.
For any integerk∈ℕ*,
(35)λk:=inf⁡K∈Γkmax⁡u∈KpΦ(u)
is a critical value of Φ restricted on 𝔐. More precisely, there exist uk∈K, such that
(36)λk=pΦ(uk)=sup⁡u∈KpΦ(u),
and uk is a solution of  (Ep)λ′ associated with positive eigenvalue λk. Moreover,
(37)λk⟶∞,ask⟶∞.Proof.
We only need to prove that, for anyk∈ℕ*, Γk≠∅ and the limit (37). Indeed, since 𝕎p(Ω) is separable, there exists (ei)i≥1 linearly dense in Wp(Ω) such that supp⁡ei∩supp⁡ej=∅ if i≠j. We may assume that ei∈𝔐 (if not, we take ei′=ei/φ(ei)).
Let nowk∈ℕ* and denote that
(38)Fk=span⁡{e1,e2,…,ek}.
Clearly, Fk is a vector subspace with dim⁡Fk=k.
Ifv∈Fk, then there exist α1,…αk in ℝ such that v=∑i=1k‍αiei. Thus
(39)φ(v)=∑i=1k‍|αi|pφ(ei)=1p∑i=1k‍|αi|p.
It follows that the map
(40)v⟼(pφ(v))1/p:=∥v∥
defines a norm on Fk. Consequently, there is a constant c>0 such that
(41)c∥Δu∥p≤∥v∥≤1c∥Δu∥p.
This implies that the set
(42)V=Fk∩{v∈𝕎p(Ω):pφ(v)≤1}
is bounded, since V⊂B(0,1/c), where
(43)B(0,1c)={u∈𝕎p(Ω),such that∥Δu∥≤1c}.
Thus, V is a symmetric bounded neighborhood of 0∈Fk. Moreover, Fk∩𝔐 is a compact set.
By Proposition  2.3 in [13], we conclude that γ(Fk∩𝔐)=k and then we obtain finally that Γk≠∅.
This completes the proof of the theorem.
Now, we claim that(44)λk⟶∞,ask⟶∞.
Let (en,ej*)n,j be a biorthogonal system such that en∈𝕎p(Ω) and ej*∈(𝕎p(Ω))′, the en are linearly dense in 𝕎p(Ω), and the ej* are total for the dual (𝕎p(Ω))′.
Fork∈ℕ*, set
(45)Fk=span⁡{e1,…,ek},Fk⊥=span⁡{ek+1,ek+2,…}.
We have, for any K∈Γk,K∩Fk⊥≠∅. Thus
(46)tk:=inf⁡K∈Γksup⁡u∈K∩Fk⊥pΦ(u)⟶∞,ask⟶∞.
Indeed, if not, for k is large, there exists uk∈Fk-1⊥ with ∥uk∥p=1 such that
(47)tk≤pΦ(uk)≤M,
for some M>0 independent of k. Thus ∥Δuk∥p≤M. This implies that (uk)k is bounded in 𝕎p(Ω). For a subsequence of {uk}, if necessary, we can assume that {uk} converges weakly in 𝕎p(Ω) and strongly in Lp(Ω). By our choice of Fk-1⊥, we have uk⇀0 weakly in 𝕎p(Ω), because 〈en*,ek〉=0, for any k≥n. This contradicts the fact that ∥uk∥p=1 for all k. Since λk≥tk the claim is proved. This achieved the proof of the theorem.Corollary 7.
One has the following statements:(i)
λ1=inf⁡{∥Δv∥pp, v∈𝕎p(Ω); ∫Γ‍|∂u/∂n|pds=1};(ii)
0<λ1≤λ2≤⋯≤λn→+∞.Proof.
(i) Foru∈𝔐, set K1={u,-u}. It is clear that γ(K1)=1, Φ is even, and
(48)pΦ(u)=max⁡K1pΦ≥inf⁡K∈Γ1max⁡KpΦ.
Hence
(49)inf⁡u∈𝔐pΦ(u)≥inf⁡K∈Γ1max⁡KpΦ=λ1.
On the other hand, for all K∈Γ1, for all u∈K, we have
(50)sup⁡KpΦ≥pΦ(u)≥inf⁡u∈𝔐pΦ(u).
It follows that
(51)inf⁡K∈Γ1max⁡KpΦ=λ1≥inf⁡u∈𝔐pΦ(u).
Thus
(52)λ1=inf⁡{∥Δv∥pp,v∈𝕎p(Ω);∫Γ‍|∂u∂n|pds=1}.
(ii) For alli≥j, we have Γi⊂Γj and in view of definition of λi,i∈ℕ*, we get λi≥λj. Regarding λn→∞, it is proved before in Theorem 6.We now turn to the fourth-order nonlinear Steklov boundary eigenvalue problem(Ep)λ.Definition 8.
One says that a functionu∈𝕎p(Ω) is weak solution of (Ep)λ if
(53)∫Ω‍|Δu|p-2ΔuΔvdx-λ∫∂Ω‍|∂u∂n|p-2∂u∂n∂v∂nds=∫Ω‍|u|p*-2uvdx∀v∈𝕎p(Ω).
If u∈𝕎p(Ω)∖{0}, then it is called an eigenfunction of problem (Ep)λ.Lemma 9 (see [19]).
Assume thatΩ is a bounded domain of C2 in ℝN.
ThenW2,p(Ω)↪Lq(Ω) for all p≤q≤Np/(N-2p).We formulate our second main result of this paper as follows.Theorem 10.
There existsλ1*<λ1 such that if λ∈(λ1*,λ1) then (Ep)λ admits at least energy solutions uλ satisfying
(54)uλ⟶0in𝕎p(Ω),uλ∥Δuλ∥p⟶u*in𝕎p(Ω)mmmmmmmmmmmmmmmmlllmmmmasλ⟶λ1,
where u* is the first eigenfunction of (Ep)λ′ such that ∥Δu*∥p=1, associated with the principal eigenvalue λ1 defined by (11).Proof of Theorem10.
Consider the following minimization problem:(55)Λλ=inf⁡u∈𝕎p(Ω)∖{0}∥Δu∥pp-λ∫Γ‍|∂u/∂n|pds∥u∥p*p.
The existence of least energy solution follows from the following proposition.Proposition 11.
Assume that0<λ<λ1. If Λλ<S, then the minimum in (55) is achieved, where S is the Sobolev constant for the embedding W2,p(Ω)↪Lp*(Ω).Proof.
Let{um}m≥0 be a minimizing sequence for Λλ such that
(56)∥um∥p*p=1.
Then,
(57)∥Δum∥pp-λ∫Γ‍|∂um∂n|pds=Λλ+o(1)(asm⟶∞).
Moreover, from (11), we have
(58)∥Δum∥pp=Λλ+λ∫Γ‍|∂um∂n|pds+o(1)≤Λλ+λλ1∥Δum∥pp+o(1),
which implies that {um} is bounded in 𝕎p(Ω). Hence {∇um} is bounded in W1,p(Ω).
Exploiting the compactness of the embedding𝕎p(Ω)↪Lp(Γ) and W1,p(Ω)↪Lp(Ω), we deduce that there exists u∈𝕎p(Ω) such that
(59)um⇀uin𝕎p(Ω),∂um∂n⟶∂u∂ninLp(Γ),um⟶uinLp(Ω),
up to a subsequence. That is, if we set vm:=um-u, then
(60)vm⇀0in𝕎p(Ω),∂vm∂n⟶0inLP(Γ),vm⟶0inLp(Ω).
On the other hand, in view of (56), we have ∥Δum∥pp≥S, so that, from (57), we obtain
(61)λ∫Γ‍|∂um∂n|pds=∥Δum∥pp-Λλ+o(1)≥S-Λλ+o(1),
which remains bounded away from 0 since Λλ<S. From this, we deduce that u≠0.
Now, In view of (59) and (60), we may rewrite (57) as
(62)∥Δu∥pp+∥Δvm∥pp-λ∫Γ‍|∂u∂n|pds=Λλ+o(1).
Moreover, by (56) and Brezis-Lieb Lemma [20], we have
(63)1=∥u+vm∥p*p*=∥u∥p*p*+∥vm∥p*p*+o(1)≤∥u∥p*p+∥vm∥p*p+o(1)≤∥u∥p*p+1S∥Δvm∥pp+o(1),
where we use also the fact that both ∥u∥p* and ∥vm∥p* do not exceed 1. Since Λλ≥0 for every 0<λ<λ1, the last inequality gives
(64)Λλ≤Λλ∥u∥p*p+ΛλS∥Δvm∥pp+o(1).
By combining this inequality with (62), we obtain
(65)∥Δu∥pp-λ∫Γ‍|∂u∂n|pds=Λλ-∥Δvm∥pp+o(1)≤Λλ∥u∥p*p+(ΛλS-1)∥Δvm∥pp+o(1)≤Λλ∥u∥p*p+o(1),
which shows that u≠0 is minimizer for (53).
Letu* be a positive eigenfunction of (Ep)λ′ and
(66)λ1*=∥Δu*∥pp-S∥u*∥p*p∥∂u*/∂n∥Lp(Γ)p.
Thus λ1*<λ1 and, for λ>λ1*, we have
(67)Λλ≤∥Δu*∥pp-λ∥∂u*/∂n∥Lp(Γ)p∥u*∥p*p<S.
We now prove the first part of (54). Indeed, in view of the characterization of u* in (53), we have
(68)Λλ≤∥Δu*∥pp-λ∥∂u*/∂n∥Lp(Γ)p∥u*∥p*p=1-(λ/λ1)∥u*∥p*p⟶0mmmmmmmmmmmmmmmmmmmmlasλ⟶λ1.
Since uλ is a least energy solution of (Ep), we have
(69)∥Δuλ∥pp-λ∥∂uλ/∂n∥Lp(Γ)p∥uλ∥p*p=Λλ.
Moreover, by taking v=uλ in (53), we get
(70)∥Δuλ∥pp-λ∥∂uλ∂n∥Lp(Γ)p=∥uλ∥p*p*.
Identities (69)-(70) readily imply that ∥uλ∥p*=Λλ(N-2p)/2p2. In turn, this and (62) show that
(71)uλ⟶0inLp*(Ω)asλ⟶λ1.
Moreover, by (69) and (68) we obtain
(72)∥Δuλ∥pp-λ∥∂uλ∂n∥Lp(Γ)p≤1-(λ/λ1)∥u*∥p*p∥uλ∥p*p.
Thus
(73)∥Δuλ∥pp≤λ∥∂uλ∂n∥Lp(Γ)p+1-(λ/λ1)∥u*∥p*p∥uλ∥p*p.
Then in view of (11), we get
(74)∥Δuλ∥pp≤λλ1∥Δuλ∥pp+1-(λ/λ1)∥u*∥p*p∥uλ∥p*p.
Hence
(75)(1-λλ1)∥Δuλ∥pp≤1-(λ/λ1)∥u*∥p*p∥uλ∥p*p.
Consequently, using (76), the last inequality implies that
(76)∥Δuλ∥pp≤∥u*∥p*-p∥uλ∥p*p⟶0asλ⟶λ1.
Finally, we conclude that
(77)uλ⟶0in𝕎p(Ω)asλ⟶λ1.
From [21], the inequality
(78)|t1-t2|p≤c{(|t1|p-2t1-|t2|p-2t2)·(t1-t2)}γ/2×(|t1|p+|t2|2)1-γ/2,
holds true for any t1,t2∈ℝ, with γ=p if 1<p<2 and γ=2 if p≥2.
By applying Hölder’s inequality, we have(79)∥Δuλ∥Δuλ∥p-Δu*∥pp=1∥Δuλ∥pP∥Δuλ-∥Δuλ∥pΔu*∥pp≤c∥Δuλ∥pp-1{G(uλ,u*)}γ/2(∥Δuλ∥pp+∥Δuλ∥pp·∥Δu*∥p)1-γ/2≤c∥Δuλ∥ppγ/2-1{G(uλ,u*)}γ/2(1+∥Δu*∥p)1-γ/2,
where
(80)G(uλ,u*)=∫Ω‍(|Δuλ|p-2Δuλ-|∥Δuλ∥pΔu*|p-2(∥Δuλ∥pΔu*))·(Δuλ-∥Δuλ∥pΔu*)dx≤∫Ω‍|Δuλ|p+∥Δuλ∥p|Δuλ|p-1|Δu*|+∥Δuλ∥pp-1|Δu*|p-1|Δuλ|+∥Δuλ∥pp|Δu*|pdx.
Then using again Hölder’s inequality we obtain
(81)G(uλ,u*)≤∥Δuλ∥pp+∥Δuλ∥pp∥Δu*∥pp+∥Δuλ∥p2p-1∥Δu*∥pp-1+∥Δuλ∥pp∥Δu*∥pp≤∥Δuλ∥pp(1+∥Δu*∥pp+∥Δuλ∥pp-1∥Δu*∥pp-1+∥Δu*∥pp).
Hence
(82)∥Δuλ∥Δuλ∥p-Δu*∥pp≤c∥Δuλ∥p(1+∥Δu*∥pp+∥Δuλ∥pp-1∥Δu*∥pp-1+∥Δu*∥pp)γ/2×(1+∥Δu*∥p)1-γ/2.
Finally, we conclude that(83)∥Δuλ∥Δuλ∥p-Δu*∥pp⟶0asλ⟶λ1
which proves (54) and then the proof of Theorem 10 is achieved.



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*Source: 498386-2014-03-19.xml*