The existence of solution for a fourth-order nonlinear partial differential equation (PDE) class involving p-biharmonic operator Δ(|Δu|p−2Δu)=λρ(x)|u|q−2uinΩ, u=Δu=0, on∂Ω, is proved by applying mountain pass theorem and a local minimization.

1. Introduction and Assumptions

The goal of this paper is to investigate the existence of nontrivial solutions for a class of nonlinear partial differential equations of the fourth order of the form Δ(|Δu|p-2Δu)=λρ|u|q-2uinΩ,u=Δu=0on∂Ω,
where Ω⊂ℝN is a smooth bounded domain in ℝN, λ∈ℝ+ is a parameter which plays the role of eigenvalue, and p and q are real numbers such that p>1 and 1<q<p**if1<p<N2,q<+∞ifp≥N2,
where p** is the critical Sobolev exponent defined by p**=Np/(N-2p) if 1<p<N/2 and +∞ if p≥N/2, and ρ is an indefinite weight in Lr(Ω) with r being so that (i)if1<p<N2,then{r>N2qfor1<q≤p,r>p**(p**-q)-1forp<q<p**,(ii)r>qifp=N2,(iii)r=1ifp>N2,
and (the Lebesgue measure) mes({x∈Ω:ρ(x)>0})≠0.Δp2u:=Δ(|Δu|p-2Δu) is an operator of the fourth order called the p-biharmonic operator. For p=2, the linear operator Δ2=ΔΔ is the iterated Laplace that multiplied with positive constant occurs often in Navier-Stokes equations as being a viscosity coefficient. Its reciprocal operator denoted (Δ2)-1 is the celebrated Green operator [1]. The boundary condition in E is of the compatibility type when p=q which can be considered as the following Hamiltonian system of the two coupled equations: -Δu=ϕp(v)inΩ,-Δv=λρϕp(u)inΩ,u=v=0
with ϕp(t)=|t|p-2t, and p′=p/(p-1) is the Hölder conjugate exponent of p, by a transformation of a problem to a known Poisson's problem and using the properties of the operator-solution stated by Agmon-Douglis-Nirenberg; see [2]. Notice that a similar system as was considered recently by [3] in the restrictive case ρ≡1. Our approach it quite different. Note that, in the case p=q, E is (p-1)-homogeneous, so we are dealing a quasilinear eigenvalue problem which is considered differently. This is few considered by [4] in the particular case ρ≡1 and Ω is smooth; and by [5, 6] with u∈W02,p(Ω) as a boundary condition, for any bounded domain.

Note also that the nonhomogeneous case is not considered there. Here we seek nontrivial solutions for E by distinguishing between two subcritical cases 1<q<p and p<q<p** which means that the critical points of the associated energy functionalA(u)=1p∫Ω|Δu|pdx-λq∫Ωρ(x)|u|qdx,are defined in X=W01,p(Ω)∩W2,p(Ω). One solution is obtained by applying classical Mountain Pass Theorem and the other solution by a local minimization technique. The restrictive case ρ≡1 and Δp2 is substituted by the well-known p-Laplacian -Δp was studied by Azorero and Alonso [7] and Elkhalil and Touzani [8] by using the fundamental properties of the first eigenvalue of the Dirichlet p-Laplacian problem.

It is important to indicate here that condition (1.2) is optimal to ensure the Palais-Smale (PS) condition is satisfied. Notice that our results are investigated without any condition on the parameter λ related to the spectrum of E (when p=q). On the other hand the condition (1.3) is assumed to have eventually nontrivial solutions.

The rest of this paper is divided in two sections as follows. In Section 2, we introduce some preliminary results and we give some technical lemmas, and in Section 3, we establish our main results.

2. Preliminaries

First, we introduce some preliminary results that we will need and some lemmas that are the key point of our results. We solve the problem E in the space X=W01,p(Ω)∩W2,p(Ω) endowed with the norm ‖u‖=(∫Ω|Δu|pdx)1/p.X equipped with this norm is an uniformly convex Banach space for 1<p<∞. Hereafter, ∥·∥p is the Lp-norm, 〈·,·〉 will denote the duality between X and its dual X′.

Definition 2.1.

u is a weak solution of E if, and only if, for all v∈W01,p(Ω)∩W2,p(Ω) we have
∫Ω|Δu|p-2ΔuΔvdx=λ∫Ωρ|u|q-2uvdx.

That is, u is a critical point of the energy functional associated to E defined on X by A(u)=1p∫Ω|Δu|pdx-λq∫Ωρ|u|qdx.

Lemma 2.2.

For any r verifying (1.2) and q satisfying (1.1) there exists a constant c=c(p,q,r)>0 such that
|∫Ωρ|u|qdx|≤c‖ρ‖r‖u‖q
and the map u→ρ|u|p-2u is strongly continuous from X into X′.

Proof.

To establish (2.4), we will divide the proof to three steps with respect to exponents p,q and N.Step 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M80"><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>N</mml:mi></mml:math></inline-formula>).

Fixing u∈W01,p(Ω)∩W2,p(Ω) and using Hölder's inequality we have
|∫ρ|u|qdx|≤‖ρ‖r‖u‖sq-1‖u‖p**,
where
q-1s=1-1r-1p**.
Such exponent s exists. Indeed: if 1<q≤p, we obtain
p-1s≥q-1s=1-1r-1p**≥1-2qN-1p**≥1-2pN-1p**=p-1p**.
Thus, it suffices to take s so that
max(1,q-1)<s<p**.
If p<q<p**, we have r>p**(p**-q)-1. So,
1r<1-qp**.
Hence,
q-1s=1-1r-1p**>q-1p**.
Therefore, it suffices to take s such that
max(1,q-1)<s<p**.

In this case, for any s>1, W01,p∩W2,p(Ω)↪Ls(Ω). Thus, for t=1/(1-(r+q′)/rq′) (some t exists because r>q=q′/(q′-1)), we get
1t+1q′+1r=1.
Hölder's inequality yields
|∫ρ|u|q|≤‖ρ‖r‖u‖t‖u‖q′.

In this case we have W01,p(Ω)∩W2,p(Ω)↪L∞(Ω) and r=1. Thus,
∫Ωρ|u|qdx≤‖u‖∞q‖ρ‖1≤c‖u‖q‖ρ‖1.
To prove the continuity of u→ρ|u|q-2u, we argue as follows.

Let (un)n≥0⊂X such that un⇀u in X. Thus, ρ|un|q-2un→ρ|u|q-2u in X′, that is, supv∈X,‖u‖≤1|∫Ωρ[|un|q-2un-|u|q-2u]vdx|=o(1).
We prove it in the case 1<p<N/2, 1<q<p** and ρ∈Lr(Ω) with r satisfying (i).

Using (2.6) or (2.10), we obtain by calculation supv∈X,‖v‖1,p≤1|∫Ωρ[|un|q-2un-|u|q-2u]vdx|≤supv∈X,‖v‖1,p≤1[‖ρ‖r‖(|un|q-2un-|u|q-2u)‖s/(q-1)‖v‖p**]≤c‖ρ‖r‖(|un|q-2un-|u|q-2u)‖s/(q-1).
Therefore, the desired result can be obtained since the limit
‖(|un|q-2un-|u|q-2u)‖s/(q-1)=o(1)
holds by using the continuity of the Nemytskii operator u→|u|q-2u from Ls(Ω)→Ls/(q-1)(Ω) and the right embedding of Sobolev space.

Remark 2.3.

If u is a solution of E associated to λ=1, then, for any α>0, α1/(p-q)u is also a weak solution of E associated to α. Hence, we can reduce the problem to λ=1.

Lemma 2.4.

For p∈(1,+∞) we have the following assertions

if 1<q<p then 𝒜 is bounded from below,

if p<q<p** thus, there exist two reel σ>0 and δ>0 such that:

From (2.18) we deduce that
A(u)≥1p‖u‖q[1-pcq‖ρ‖r‖u‖q-p].
If ∥u∥≤((q/pc)(1/∥ρ∥r))1/(q-p), then there exists δ>0 such that
A(u)>δ.
If ∥u∥=((q/pc)(1/∥ρ∥r))1/(q-p), we get
A(u)>0.
The estimation above completes the proof.

Proposition 2.5.

If the pair (p,q) satisfies (1.1) with p≠q, then 𝒜 satisfies the (PS) condition.

Proof.

Let (uj)j be a sequence of Palais-Smale of 𝒜 in X. Thus there exists M>0 such that |A(uj)|≤M
and for any ɛ>0 there is j0>0 such that |〈A′(uj),uj〉|≤ɛ‖uj‖,∀j≥j0.
Claim that (uj)j is a bounded sequence in X.Step 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M169"><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>q</mml:mi></mml:math></inline-formula>).

By (2.18) we deduce that 𝒜 is coercive, then the claim follows.

Equation (PS)2 implies that, for all j>j0,
-ɛ‖uj‖≤〈A′(uj),uj〉≤ɛ‖uj‖.
Thanks to (PS)1, we obtain,
-M≤1p∫Ω|Δuj|pdx-1q∫ρ|uj|qdx≤M.
Multiplying (2.24) by (-1/q) and adding with (2.18), we obtain,
(1p-1q)‖uj‖p≤M+ɛq‖uj‖.
Hence, there exists a positive constant c>0 such that
‖uj‖≤c∀j≥j0.
So (uj) is bounded in the two cases in X and the claim is concluded. Consequently, there exists a subsequence still denoted (uj)j converges weakly for some u∈X and strongly in Lp(Ω) and in Lq(Ω) for all (p,q) satisfies (1.2).

Let 〈Ju,v〉=∫Ω|Δu|p-2ΔuΔvdx.
Thus
〈A′(uj)-A′(u),uj-u〉+∫Ωρ(|uj|q-2uj-|u|q-2u)(uj-u)=〈Juj-Ju,uj-u〉.
Since u→ρ|u|q-2u is strongly continuous from X into X′, we deduce from (PS)2, as j→+∞ in (2.29), that
limj→+∞〈Juj-Ju,uj-u〉=0.
On the other hand, we have
〈Juj-Ju,uj-u〉≥(‖Δuj‖pp-1-‖Δu‖pp-1)(‖Δuj‖p-‖Δu‖p).≥0.
This and (2.30) imply that
‖Δuj‖p⟶‖Δu‖pasj⟶+∞.
Finally, since W01,p(Ω)∩W2,p(Ω) is uniformly convex, the proof is achieved.

3. Main Results3.1. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M199"><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>q</mml:mi><mml:mo><</mml:mo><mml:mi>p</mml:mi></mml:math></inline-formula>: Local Minimization Technique

In this subsection, we show that the problem E has a sequence of weak solution by using the results of Lusternik-Shnirleman [9]. In other words, we use a local minimization for the corresponding energy functional.

Let Γn={B⊂W01,p(Ω)∩W2,p(Ω),compactsymmetricandγ(B)≥n} and set cn=infB∈Γn(supu∈BA(u)),∀n∈N*.
We will show that the sequence cn, defined by (3.1) is critical values of 𝒜. Here and in the following γ(B)=k is the genus of B, that is, the smallest integer k such that there is an odd continuous map from B into ℝk∖{0}.

Lemma 3.1.

For any k∈ℕ*,
Γk≠∅.

Proof.

W01,p(Ω)∩W2,p(Ω) is separable. Therefore, for any k∈ℕ*, there exists a finite sequence of functions u1,…,uk in W01,p(Ω)∩W2,p(Ω) linearly independent such that suppui∩suppuj=∅ for i≠j and thanks to (1.3) we can choose ui such that ∫Ωg|ui|qdx=1. Let Fk=span(u1,…,uk) be a subspace in W01,p(Ω)∩W2,p(Ω) spend by ui of dimension k.

If v∈Fk, then, there exist β1,…,βk real numbers such that v=∑i=1i=kβiui.

Thus, ∫Ωρ|v|qdx=∑i=1i=k|βi|q. Hence, the mapv⟶(∫Ωρ|u|qdx)1/q
is a norm in Fk. Consequently, there exists a constant c>0 such that
c‖u‖≤(∫ρ|u|qdx)1/q≤1c‖u‖,∀v∈Fk.
Set
Uk=Fk∩{v∈X:θ2≤∫Ωρ|v|qdx≤θ}
with θ=(p/qcp)q/(p-1). It is clear that Uk is a closed neighborhood, symmetric, compact not containing 0. Finally, by the property of genus we get γ(Uk)=k and Γk≠∅.

Theorem 3.2.

Let cn be a sequence defined by (3.1). Then, we have the following statements:

For any n∈ℕ*, there exist n distinct pairs critical points of the functional 𝒜.

If
-∞<c=cn=cn+1=⋯=cn+k
then
γ(Kc)≥k+1,
where
Kc={u∈W01,p(Ω)∩W2,p(Ω)suchthatA′(u)=0,A(u)=c}.

Proof.

From the result of [10], it suffices to prove (i).

Recall that the functional A is even, bounded from below and supXA(u)<0.

In view of Lemma 2.4, A is of class C1 on X; satisfying the (PS) condition.

Hence, these properties confirm the hypotheses of Clark's Lemma cf. [11] which proves (i).

Remark 3.3.

The set K={u∈W01,p(Ω)∩W1,p(Ω):𝒜′(u)=0} is compact. Indeed, let (un)n be a sequence in K, that is,
A′(un)=0,∀n∈N.
It is clear that, for p>q,
A(un)≤〈A′(un),un〉=0,∀n.
Since A is bounded from below, by Lemma 2.4, (un)n is a sequence of Palais-Smale (PS). Thus, (un)n possesses a convergent subsequence.

Consequently, K is compact in W01,p(Ω)∩W2,p(Ω) and the functional 𝒜 has an infinity of critical points.

3.2. The Case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M263"><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>q</mml:mi><mml:mo><</mml:mo><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi><mml:mi>*</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>: Mountain Pass Theorem

Here, we prove the existence of solutions of problem E, by using Mountain Pass theorem [11].

Lemma 3.4.

Suppose that p<q<p** and ρ∈Lr(Ω) with r satisfying (1.2). Hence there exists e∈X∖{0} such that
A(e)≤0.

Proof.

We have for any u∈X and t∈ℝ,
A(tu)=tpp∫Ω|Δu|pdx-tqq∫Ωρ|u|qdx.
From (1.3), there exists u0∈X such that ∫Ωρ|u0|qdx=1. Thus,
limt→+∞A(tu0)=-∞,becausep<q.
Hence, there is t0>0 large enough so that 𝒜(t0u0)<0. To achieve the proof, it suffices to take e=t0u0.

Consequently, we conclude the following statements.

𝒜 is C1 of class C1, even and 𝒜(0)=0.

𝒜 satisfies the (PS) condition.

There exist two positive reel l>0, δ>0 such that
A(u)>δif0<‖u‖<l,A(u)≥0if‖u‖=l.

There exists e∈X∖{0} such that 𝒜(e)≤0.

Now, we can establish the following theorem.

Theorem 3.5.

If p<q<p**, then the value c defined by
c=infγ∈Gmaxt∈[0,1]A(γ(t))
is a critical value of 𝒜.

In view of Lemma 2.4., Proposition 2.5, and properties (a)–(d), c is a critical value of A by applying the Mountain Pass Theorem due to [11].

Acknowledgment

The author gratefully acknowledges the financial support provided by Al-Imam Muhammed Ibn Saud Islamic University during this research.

LionsJ. L.GilbargD.TrudingerN. S.Marcos do ÓJ.UbillaP.A multiplicity result for a class of superquadratic Hamiltonian systemsDrábekP.ÔtaniM.Global bifurcation result for the p-biharmonic operatorEl khalilA.KelatiS.TouzaniA.On the spectrum of the p-biharmonic operatorEl khalilA.KelatiS.TouzaniA.A nonlinear boundary problem involving the p-biharmonic operatorGarcía AzoreroJ. P.Peral AlonsoI.Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvaluesElkhalilA.TouzaniA.On the first eigencurve of the p-LaplacianSzulkinA.Ljusternik-Schnirelmann theory on C1-manifoldsCostaD. G.Topicos en análise náo e lineare eplioes s equaoes diferenciaisProceedings of ELAM VIII1986Rio de Janeiro, BrazilSpriger Lecture NotesAmbrosettiA.RabinowitzP. H.Dual variational methods in critical point theory and applications