Existence of Solution for Nonlocal Heterogeneous Elliptic Problems
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Existence of Solution for Nonlocal Heterogeneous Elliptic Problems Mahmoud Bousselsal and Elmehdi Zaouche Abstract. We consider a class of a nonlocal heterogeneous elliptic problem of type −M (|u|qq ) div(a(x)∇u) = g(x, u) with a homogeneous Dirichlet boundary condition. Under different assumptions on the function g, we establish two existence theorems for this problem by using, respectively, the Schauder and Tychonoff fixed point theorems. Also, we give an example for each theorem. Mathematics Subject Classification. 35J60, 35J75, 47H10. Keywords. Nonlocal heterogeneous elliptic problem, Homogeneous Dirichlet boundary condition, Schauder and Tychonoff fixed point theorems, Existence.
1. Introduction Let Ω be a bounded domain in Rn (n ≥ 1). We consider the following weak formulation of the nonlocal heterogeneous elliptic problem ⎧ Find u ∈ H01 (Ω) such that : ⎪ ⎪ ⎪ ⎨ M (|u|qq ) a(x)∇u · ∇ξ dx = g(x, u)ξ dx (P) ⎪ Ω Ω ⎪ ⎪ ⎩ ∀ξ ∈ H01 (Ω), where 1 ≤ q ≤ 2, for a.e. x ∈ Ω, a(x) = (aij (x))ij is an n×n matrix satisfying for two positive constants λ, Λ: ∀ξ ∈ Rn : n
∀ξ ∈ R :
λ|ξ|2 ≤ a(x)ξ · ξ |a(x)ξ| ≤ Λ|ξ|
a.e. x ∈ Ω, a.e. x ∈ Ω,
(1.1) (1.2)
and M : R → R is a continuous function such that, for some constant m0 > 0, we have M (t) ≥ m0
∀t ∈ R.
0123456789().: V,-vol
(1.3)
129
Page 2 of 10
M. Bousselsal and E. Zaouche
MJOM
Under different assumptions on the function g, we prove two existence theorems for the problem (P ) by using, respectively, the Schauder and Tychonoff fixed point theorems. Throughout the first theorem, we assume that g : Ω × R → R is a function satisfying x → g(x, t) is measurable for all t ∈ R, g(·, 0) ∈ L2 (Ω), g(·, 0) = 0 a.e. in Ω and b(x)|t − s|p , 0 < p < 1; ∀t, s ∈ R, a.e. x ∈ Ω, |g(x, t) − g(x, s)| ≤ C|t − s|, p = 1, (1.4) 2 1−p
0 where b ∈ L (Ω) and C is a positive constant such that C < λm 2 , with CΩ 1 CΩ is a Poincar´e constant for H0 (Ω). In the second theorem, x → g(x, t) is measurable for all t ∈ R, t → g(x, t) is continuous on R for a.e. x ∈ Ω, g(·, 0) = 0 a.e. in Ω and there exists h ∈ L2 (Ω) such that
∀t ∈ R, a.e. x ∈ Ω,
|g(x, t)| ≤ h(x).
(1.5)
1 We note that, if Ω is of class C , we can assume in both theorems that 2n q ∈ [1, +∞) and q ∈ 1, n−2 , respectively, in the cases n = 2 and n > 2, since the embedding H 1 (Ω) → Lq (Ω) is compact, by applying Tychonoff’s fixed point theorem. In [1,3–8,14], the authors studied classes of nonlocal problems motivated by the fact that these classes arise in one of the physical models. For instance, if q = 2, we can employ the problem (P ) to study some questions related to nonlinear deflection of beams from the well-known Carrier’s Equation. For more details in the physical background, we refer to [4–6,8,16]. The problem (P ) was studied in [8] for a(x) = In , where In is the identity matrix in Rn×n . Under two conditions of invertibility and boundedness on the function M and g(x, t) = α(x) + β(x) depends only on x, with α, β ∈ C(Ω), α, β > 0 in Ω, an existence theorem of a positive solution was obtained by using a result from the
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