On a Nonlinear PDE Involving Weighted p-Laplacian

on a ball Br ⊂ R (N ≥ 2). Under some appropriate conditions on the functions f,w and the nonlinearity 1 (1−u) , we prove the existence and the uniqueness of solutions of the above problem. Our analysis mainly combines the variational method and critical point theory. Such solution is obtained as a minimizer for the energy functional associated with our problem in the setting of the weighted Sobolev spaces.


Introduction
Differential equations and variational problems have many applications in mathematical physics such as in the Micro-Electro Mechanical Systems(MEMS), 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 an elastic beam, see for instance ( [10], [11]) or [16] and the references therein.
The most important linear partial differential equations of the second odrer are governed by the celebrated Laplacian operator ∆.It is less well-known that is also a nonlinear counterpart, the so called p-Laplacian defined by ∆ p u = ∇(|∇u| p−2 u).At the critical points (∇u = 0), this prototype of nonlinear operator for p = 2 is degenerate for p > 2 and singular for p < 2. For p = 2, we just get the usual Laplacian operator.During the last quarter of cycle the Partial differential 132 A. El Khalil, M.D. Morchid Alaoui, M. Laghzal and A. Touzani equations governed by p-Laplacian have been much studied and its theory is by now rather developed.The purpose of this paper is to established the existence and uniqueness of the solutions for the following nonlinear elliptic equation with the weighted p-Laplacian operator where B r is an open ball in R N (N ≥ 2), of radius r > 0 and centered at the origin, 1 < p < ∞, w is a positive weight function locally integrable in R N , i.e., w ∈ L 1 loc (R N )), f is a positive nonzero bounded continuous function.Notice that the nonlinearity Our methode is more direct and is mainly based on the critical point theory.For degenerate partial differential equations, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces ( [3], [4], [5], [8], [15]).The type of a weight depends on the equation type.
A class of weights, which is particularly well understood, is the class of A pweights (or Muckenhoupt class) that was introduced by Muckenhoupt [8].These classes have found many useful applications in harmonic analysis in the linear case [13] and [14].Another reason for studying A p -weights is the fact that powers of the distance to submanifolds of R N often belong to A p [1] and [15].There are, in fact, many interesting examples of weights [5].
In the particular case p = 2 with w(x) ≡ 1 and [2] studied the problem: in Ω, The authors established the existence of a regular as well as a singular solution to (1.2).This simple model, which lends itself to the vast literature on second order quasi-linear eigenvalue problems, is already a rich source of interesting mathematical problems.
In the degenerate case, the weighted p-Laplacian operator has been studied [1] and references cited there in.
Inspired by the above-mentioned papers, we study the existence and uniqueness of solutions of problem (1.1) in a neighborhood of the origin.More precisely, under some appropriate conditions on the functions f, w and the nonlinearity 1  (1−u) 2 .The paper is organized as follow.First, in Section 2, we recall and we prove some preliminary results which will be used later.In Section 3, we establish the existence and uniqueness of solutions for problem (1.1).Finally, the last Section, we give an application illustrating our main results.
On a Nonlinear PDE Involving Weighted p-Laplacian 133

Preliminary Results
Before we discuss some results concerning the problem (1.1), let us recall some various definitions and basic properties of the weighted Sobolev spaces.
For convenience, let both dx and |.| stand for the( N -dimensional) Lebesgue measure in R N .As we shall always a positive weight a locally integrable function on R N .Every weight w gives rise to a measure on the measurable subsets of R N through integration.This measure will be denoted by µ, That is:

Muckenhoupt weights.
We briefly recall some fundamentals on Muckenhoupt classes A p .

Definition 2.1. Let w be a positive, locally integrable function on R
Remark 2.2.Since the pioneering works of Muckenhoupt [7] and [9] these classes of weight functions have been studied in great detail.In present paper, we are only concerned with the case p > 1.

Definition 2.3. (space of functions of bounded mean oscillation (BMO)).
Suppose that f is integrable over compact sets in R N and that for any ball B ⊂ R N , with volume denoted by |B|, the mean of f over B will be We say that f belongs to BMO if where the supremum is taken over all balls B. Here, ||f || * is called the BMO-norm of f , and it becomes a norm on BMO after dividing out the constant functions.
Remark 2.4.(i) Functions of bounded mean oscillation were introduced by F. John and L. Nirenberg [6] and Simon [12].

Example 2.5. (i). One of the most prominent examples of an
given by [14].An other example is given by w(x) = e αϕ(x) which belongs in A 2 , whenever ϕ ∈ BM O(R N ) and the real α > 0 [12].
Lemma 2.6.Let w ∈ A p ,where 1 < p < ∞,and let E be a measurable subsets of a ball B. then Proof: By writing 1 = w 1/p w −1/p ,Hölder's inequality implies that Note that the dual space of L p (Ω, w) is the space [L p (Ω, w)] * = L p ′ (Ω, w * ) where w * = w −1/p−1 and the conjugate index of p will be denoted by p ′ in such a way that (1/p + 1/p ′ = 1) .Lemma 2.9.[15] If w ∈ A p , 1 < p < ∞, then since w −1/(p−1) is locally integrable, we have L p (Ω, w) ⊂ L 1 loc (Ω).Proof: Suppose that f ∈ L p (Ω, w), and let B ⊂ Ω be a ball.Thus Hölder's inequality implies that, endowed by the weighted norm We also define the space W k,p 0 (Ω, w) as the closure in W k,p (Ω, w) of C ∞ 0 (Ω) with respect to the norm ||.|| k,p,Ω .Proposition 2.12.Let Ω ⊂ R N be a bounded open set, 1 < p < ∞, k be a nonnegative integer, suppose that w ∈ A p .Then The spaces W k,p (Ω, w) and W k,p 0 (Ω, w) are Banach spaces.Remark 2.13.It is evident that a weight function w which satisfies 0 < C 1 ≤ w(x) ≤ C 2 ,for x ∈ Ω, gives nothing new (the space W k,p (Ω, w))is then identical with the classical Sobolev space W k,p (Ω).Consequently, we shall be interested above all in such weight functions w which either vanish somewhere in Ω ∪ ∂Ω or increase to infinity (or both).
In order to avoid too many suffices, at each step, a generic constant is denoted by C Br or C. we need the following basic result.Then, endowed with ||.|| X , X is a separable and reflexive Banach space.

Main Result
We need the following assumption.
x ∈ B r , w(x) > 0 and f (x) > 0, we have Then, for w ∈ A p (with 1 < p < ∞), we have also (b) One of the most prominent examples of f (x), is given by the function where α ≥ 0 and β > 0.
Now, we are ready to state our main theorem of this paper.
In order to prove Theorem 3.3, we need the following auxillary lemma.
Lemma 3.4.We have the following statements (a) Ψ is weakly lower semi-continuous,Ψ ∈ C 1 (X, R),and Proof: We start first by showing that Ψ ∈ C 1 (X, R), that is, for all h ∈ X, and dΨ : X → X * is continuous, where we denote by X * the dual space of X.For all h ∈ X,we have Using condition (H) and Hölder's inequality, we obtain Using the linearity of dΨ(u) and the above inequality, we deduce that dΨ(u) ∈ X * .Note that the function u By the continuity and the convexity of Ψ, we deduce that Ψ is weakly lower semicontinuous .(b) Similarly, we can also prove that Φ ∈ C 1 (X, R).Moreover since 1 < p < ∞ and in view of the weakly lower semi-continuity of the norm , we deduce that Φ is lower semi-continuous for the weak convergence.Furthermore, for all u, v ∈ X.Which gives the Fréchet differentiability of Φ. ✷ Proof: of Theorem 3.3 Our aim is to obtain a minimizer as the limit of a minimizing sequence {u n } of the Euler-Lagrange functional I p , which is a weak solution of problem (1.1).
We will divide the proof into five steps.
Step 1.We shall prove that inf{I(u)|u ∈ X} > −∞.Bearing in mind our definition of norm, then (3.2) implies Young's inequality yields On the other hand, we have In view of (3.3) and (3.4), we deduce that that is, I is bounded from below.This completes step 1.
Step 2. We shall prove that any minimizing sequence is bounded in X Let {u n } be a minimizing sequence, that is, a sequence such that and satisfies (H).Then for n large enough, we obtain that and we get by applying Theorem 2.14 Hence u n is bounded in X.By the reflexivity of the space X , there exists a function u ∈ X such that u n ⇀ u in X (for a subsequence if necessary).
Step 3. We shall prove that I is weakly lower semi-continuous.By (a) and (b) of Lemma 3.4, I is weakly lower semi-continuous, I ∈ C 1 (X, R).It follows that and thus u is a minimizer of I on X.
Step 4. We shall prove that u is a minimizer of I and it's equivalently a weak solution of problem (1.1).
For any ϕ ∈ X, the function has a minimum at θ = 0. Hence  In other words u is a weak solution of problem (1.1).
Step 5. We claim that the limit function u is unique.
We shall prove that Φ ′ is strictly monotone.Let u, v ∈ X are two weak solutions of problem (1.1), with u = v in X.
We recall the following well-known inequalities [12], which hold for every a, b ∈ R N
Remark 2.10.(i) if Ω is bounded,one obtains in the same way that L p (Ω, w) is continuously embedded in L 1 (Ω).
p (Ω, w) .Definition 2.11.Let Ω ⊂ R N be a bounded open set, for 1 < p < ∞, w ∈ A p and a positive integer k, the weighted Sobolev spaces W k,p (Ω, w) is defined by