Existence of Solutions to Elliptic Equations on Compact Riemannian Manifolds

: The aim of this paper is to investigate the existence of weak solutions of a nonlinear elliptic problem with Dirichlet boundary value condition, in the framework of Sobolev spaces on compact Riemannian manifolds.


Introduction
In this paper, we study the following nonlinear elliptic problem (P )    −div g (a (x, u, ∇u)) + b (x, u, ∇u) + λ|u| q−2 u + h(x)|u| r−2 u = f (x, u) in M, u = 0 on Γ, (M, g) is a smooth compact Riemannian manifold of dimension N , h(x) ∈ L s (M ) with s = q q−r , λ > 0, 1 < r < p < q < p * , here p * = N p N −p if p < N or p * = ∞ if p ≥ N , and where Γ i ij represents the symbol of Christoffel and a is a Carathtéodory's function defined from M × R × R N into R, (measurable with respect to x in M for every (s, η) in R × R N and continuous with respect to (s, η) in R × R N for almost every x in M ), and satisfy the assumptions of growth, ellipticity and monotonicity.b: M × R × R N → R is a Carathéodory's function that satisfy a few conditions that we will review in the section 3, and f : M × R → R is a Carathéodory's function which is decreasing with respect to the second variable and satisfy the growth assumption.Boccardo and Gallouët [11] considered the following problem − div (a(x, u, ∇u)) = f (x, u, ∇u) in Ω u = 0 on ∂Ω, (1.1) where the right-hand side f is a bounded Radon measure, they proved the existence and some regularity results.In [13], Duc and Vu showed the existence of weak solutions for the problem (1.1), by using a variation of the Mountain Pass theorem which was introduced by Duc in [14].Moreover, we can even cite the work of S. Liu [23], he established the existence of weak solutions to the particular case for the problem (1.1) by using the Morse theory.Bensoussan et al. [7] proved the existence of a solution for the problem − div (a(x, u, ∇u)) where f ∈ W −1,p′ (Ω).Drabek and Nicolosi in [12] proved the existence of bounded solution for the degenerated problem (1.2) where g(x, u, ∇u) = −c 0 |u| p−2 u.
A abbassi et al. [1] established the existence results of weak solutions via the recent Berkovits topological degree for the following nonlinear p-elliptic problems : where the vector field f is a Carathéodory function which satisfies only the growth condition.
In [6], the authors studied the following problem by using Browder's theorem, they proved the existence and uniqueness of a weak solution.When p = 2, the problem (1.4) is a normal Schrodinger equation which has been extensively studied.
The Sobolev space on Riemannian manifolds is another area that is rapidly developing.Thierry Aubin's work in 1976 is credited with providing the first understanding of the Sobolev spaces on the Riemannian manifolds (See [2,3,4,5,8,9,16,17,18,21,22,23]).He applied his findings to the non-linear EDPs on the manifolds.Sobolev spaces on compact manifolds have been used for a long time (Ebin works); in essence, they are the same as Sobolev spaces on a ball of R n .This paper is organized as follows.In Section 2, we recall some preliminaries about the framework of Sobolev space on the Riemannian manifolds and some technical lemmas.In Section 3, we introduce some assumptions on the Carathéodory functions a i (x, s, ξ), b(x, s, ξ) and f (x, s) for which our problem has at least one solution and we prove our main result.

Preliminaries
In this section, we recall the most important and relevant properties and notations about Sobolev spaces on the Riemannian manifolds, and we give some properties and lemmas, that we will need in our analysis of the problem (P ), by that, referring to [15,16,17] for more details.

Definitions and properties
Let (M, g) a Riemannian manifold, and dσ g the Riemannian measure associated with it.Given u : M → R a function of class C ∞ (M ), and k an integer, we denote by ∇ k u the k th covariant derivative of u (with the Convention ∇ 0 u = u ) and |∇ k (u)| the norm of ∇ k u defined by where the Einstein summons is adopted.Let p ≥ 1 be a real number, and k is a positive integer.We define the following spaces: Existence of Solutions to Elliptic Equations 3 where u 1,p = ∇u p + u p .
Definition 2.2.We must recall the notion of the geodesic distance for every curve: We define the length of Υ by: Remark 2.3.For x, y ∈ M defining a distance d g by: By the theorem of Hopf-Rinow, we obtain that if M a Riemannian manifold then compact for all x, y in M can be joined by a minimizing curve Υ i.e l(Υ) = d g (x, y).
Proposition 2.4.If p = 2, the space W k,2 (M ) is a Hilbert space for the following scalar product
Corollary 2.11.Let M be a compact Riemannian manifold of n dimension.Assume that is a continuous and compact embedding.
Lemma 2.12.(Inequality of Poincaré)( [17].)Let D a regular domain is bounded in a Riemannian manifold M and 1 ≤ p < ∞.Then there is a constant A such as: By combining this lemma with the Hölder inequality, we obtain: Proof.We apply the inequality of Hölder, we will have .

Technical lemmas
Theorem 2.14.( [10].)Let X be a Banach Reflexive space and let A : X → X ′ an operator having the following properties: (1) A is bounded hemicontinuous .
(3) A is coercive ,e.i, Then A is surjective of X → X ′ , i.e, for every f ∈ X ′ , there exists u ∈ X such as: The next inequalities will be systematically used in this work.

Existence Result
In this section, we will state and prove the existence of solutions for problem (P ).We start by stating the following assumptions : The function a i (x, η, ξ) is a Carathéodory function satisfies the following assumptions: • For a.e x ∈ M and all (η, ξ • For a.e x ∈ M , for all (η, ξ) ∈ R × R N , C 0 > 0 and (H 3 ) b is a Carathéodory function satisfies the following assumptions: • For a.e x ∈ M and for all (s, ξ • For a.e x ∈ M , for all (ξ (H 4 ) f Carathéodory function which is decreasing with respect to the second variable, i.e • For a.e x ∈ M , for all ξ ∈ R, C > 0 and f 0 ∈ L q′ (M ); (3.9) Definition 3.1.We say that u ∈ W 1,p 0 (M ) is a weak solution of Dirichlet problem (P ) if Theorem 3.2.Let (M, g) be a compact Riemannian manifold of N dimension, suppose that the hypotheses (H 1 ) − (H 4 ) are satisfied.Then the problem (P ) admits at least a weak solution u ∈ W 1,p 0 (M ).
Proof.We define the operator T : W 1,p 0 (M ) → W −1,p′ (M ) by where the operators A,B, G, R and F are defined from W 1,p 0 (M ) into W −1,p′ (M ) as for all u, v ∈ W 1,p 0 (M ).By Definition 3.1, the main tool in searching the weak solutions of (P ) is to finding u ∈ W 1,p 0 (M ) which satisfies the operator equation T u = 0.
Step 1 : We establish that T is bounded operator.Let u ∈ W 1,p 0 (M ), such that u 1,p ≤ M .Using Hölder's inequality, we obtain , by the growth condition (3.2) and the continuous embedding W 1,p 0 (M ) ֒→ L q (M ), we have Existence of Solutions to Elliptic Equations 7 Hence, A is bounded.
For each u ∈ W 1,p 0 (M ), we have by the growth condition (3.5), inequality of Hölder and the continuous embedding W 1,p 0 (M ) ֒→ L q (M ) that Hence, the operator B is bounded.
For each u ∈ W 1,p 0 (M ), by using the inequality of Hölder and the continuous embedding and where m = q r−1 and 1 s + 1 m + 1 q = 1.Hence, G and R are bounded.For each u ∈ W 1,p 0 (M ), we have by the growth condition (3.9), inequality of Hölder and the continuous embedding Hence, F is bounded.
Step 2 : We establish that T is continuous. and Hence there exist a subsequence (u n ) and measurable functions h in L q (M ) and g in (L p (M )) N such that for a.e.x ∈ M and for all k ∈ N. Since and taking into account the inequality , the dominated convergence theorem imply that Au n → Au in W −1,p′ (M ).Then A is continuous. Similarly, Then B,R, G and F are continuous.
Step 3 : We establish that T is monotone.We have By Lemma 2.15, we have Similarly, by Lemma 2.15 and condition (3.1), we have On the other hand, Therefore T is monotone.
Step 4 : We establish that T is a coercive operator.
Using the growth condition (3.9), inequality of Hölder and the continuous embedding W 1,p 0 (M ) ֒→ L q (M ), we have ≥ C 2 u p 1,p − C 1 + λ u q q + h 0 u r r − C ′ u 1,p .
which gives directly the coercivity of the operator T .By applying Theorem 2.14 we deduce that the problem (P ) admits at least one solution u ∈ W 1,p 0 (M ).