On a Class of PDE Involving p-Biharmonic Operator

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.


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 where Ω ⊂ R N is a smooth bounded domain in R N , λ ∈ R is a parameter which plays the role of eigenvalue, and p and q are real numbers such that p > 1 and 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 L r Ω with r being so that for 1 < q ≤ p, r > p * * p * * − q −1 for p < q < p * * , 1.2 and the Lebesgue measure mes x ∈ Ω : ρ x > 0 / 0. 1.3 Δu is an operator of the fourth order called the p-biharmonic operator.For p 2, the linear operator Δ 2 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 Δ 2 −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: with φ p t |t| p−2 t, and p p/ p − 1 is the H ölder 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 ∈ W 2,p 0 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 functional p is substituted by the well-known p-Laplacian −Δ p 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.

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 W X equipped with this norm is an uniformly convex Banach space for 1 < p < ∞.Hereafter, • p is the L p -norm, •, • will denote the duality between X and its dual X .
That is, u is a critical point of the energy functional associated to E defined on X by 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 Ω ρ|u| q dx ≤ c ρ r u q 2.4 and the map u → ρ|u| p−2 u 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.

ISRN Mathematical Analysis
Step 1 1 < p < N .Fixing u ∈ W 1,p 0 Ω ∩ W 2,p Ω and using H ölder's inequality we have where Such exponent s exists.Indeed: if 1 < q ≤ p, we obtain

2.13
Step 3 p > N/2 .In this case we have W

2.14
To prove the continuity of u → ρ|u| q−2 u, we argue as follows. Let

2.15
We prove it in the case 1 < p < N/2, 1 < q < p * * and ρ ∈ L r Ω with r satisfying i .Using 2.6 or 2.10 , we obtain by calculation

2.16
Therefore, the desired result can be obtained since the limit holds by using the continuity of the Nemytskii operator u → |u| q−2 u from L s Ω → L s/ 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, α 1/ 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 ii if p < q < p * * thus, there exist two reel σ > 0 and δ > 0 such that: On the other hand, we have ≥ 0.

The Case 1 < q < p: 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.We will show that the sequence c n , 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 p Ω is separable.Therefore, for any k ∈ N * , there exists a finite sequence of functions u 1 , . . ., u k in W 1,p 0 Ω ∩W 2,p Ω linearly independent such that supp u i ∩supp u j ∅ for i / j and thanks to 1.3 we can choose u i such that with θ p/qc p q/ p−1 .It is clear that U k is a closed neighborhood, symmetric, compact not containing 0. Finally, by the property of genus we get γ U k k and Γ k / ∅.Theorem 3.2.Let c n be a sequence defined by 3.1 .Then, we have the following statements: i For any n ∈ N * , there exist n distinct pairs critical points of the functional A.
ii If where Proof.From the result of 10 , it suffices to prove i .Recall that the functional A is even, bounded from below and sup X A u < 0. In view of Lemma 2.4, A is of class C 1 on X; satisfying the PS condition.Hence, these properties confirm the hypotheses of Clark's Lemma cf.11 which proves i .
It is clear that, for p > q, A u n ≤ A u n , u n 0, ∀n.

3.10
Since A is bounded from below, by Lemma 2.4, u n n is a sequence of Palais-Smale PS .Thus, u n n possesses a convergent subsequence.Consequently, K is compact in W 1,p 0 Ω ∩ W 2,p Ω and the functional A has an infinity of critical points.

The Case p < q < p * * : Mountain Pass Theorem
Here, we prove the existence of solutions of problem E , by using Mountain Pass theorem 11 .

3.13
Hence, there is t 0 > 0 large enough so that A t 0 u 0 < 0. To achieve the proof, it suffices to take e t 0 u 0 .

3.16
Proof.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 .
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 Δ 2 symmetric and γ B ≥ n} and set c n inf B∈Γ n sup u∈B A u , ∀n ∈ N * .3.1

aTheorem 3 . 5 .
A is C 1 of class C 1 , even and A 0 0. b A satisfies the PS condition.cThere exist two positive reel l > 0, δ > 0 such thatA u > δ if 0 < u < l, A u ≥ 0 if u l.3.14 d There exists e ∈ X \ {0} such that A e ≤ 0. Now, we can establish the following theorem.If p < q < p * * , then the value c defined by