Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations
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Research Article Entire Bounded Solutions for a Class of Quasilinear Elliptic Equations Zuodong Yang and Bing Xu Received 29 June 2006; Accepted 17 October 2006 Recommended by Shujie Li
We consider the problem −div(|∇u| p−2 ∇u) = a(x)(um + λun ),x ∈ RN ,N ≥ 3, where 0 < m < p − 1 < n,a(x) ≥ 0,a(x) is not identically zero. Under the condition that a(x) satisfies (H), we show that there exists λ0 > 0 such that the above-mentioned equation admits at least one solution for all λ ∈ (0,λ0 ). This extends the results of Laplace equation to the case of p-Laplace equation. Copyright © 2007 Z. Yang and B. Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, we are interested in studying the existence of solutions to the following quasilinear equation: −div |∇u| p−2 ∇u = a(x) um + λun ,
x ∈ RN , N ≥ 3,
(1)
where 0 < m < p − 1 < n, a(x) ≥ 0, a(x) is not identically zero. We will assume throughout the paper that a(x) ∈ C(RN ). Equations of the above form are mathematical models occuring in studies of the p-Laplace equation, generalized reaction-diffusion theory [1], non-Newtonian fluid theory, and the turbulent flow of a gas in porous medium [2]. In the non-Newtonian fluid theory, the quantity p is characteristic of the medium. Media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics. If p = 2, they are Newtonian fluids. Problem (1) for bounded domains with zero Dirichlet condition has been extensively studied (even for more general sublinear functions). We refer in particular to [3–10] (see also the references therein). When p = 2, the related results have been obtained by [11– 16] (including bounded domains with zero Dirichlet condition or RN ). Our existence
2
Boundary Value Problems
results extend that of Brezis and Kamin (see [11, Theorem 1]) for semilinear problem, and complement results in [3–10]. u ∈ W 1,p (RN ) ∩ C 1 (RN ) is called a entire weak solution to (1) if
RN
|∇u| p−2 ∇u · ∇ψ dx =
a(x) um + λun ψ dx
RN
∀ψ ∈ C0∞ RN
(2)
and u > 0 in RN . Definition 1. u ∈ W 1,p (RN ) ∩ C 1 (RN ) is called a supersolution to problem
div |∇u| p−2 ∇u + f (x,u) = 0 if
RN
|∇u| p−2 ∇u · ∇ψ dx ≥
∀ψ ∈ C0∞ RN
f (x,u)ψ dx
RN
(3)
(4)
and u > 0 in RN . As always, a subsolution u is defined by reversing the inequalities. From [3], we have the following lemma. Lemma 1. Suppose that f (x,u) is defined on RN+1 and is locally H¨older continuous (with exponent λ ∈ (0,1)) in x. u is a subsolution and u is a supersolution to (3) with u ≤ u on RN , and suppose that f (x,u) is locally Lipschitz continuous in u on the set
(x,u) : x ∈ RN , w(x) ≤ u ≤ v(x) .
(5)
Then, (3) possesses an entire solution u(x) satisfying x ∈ RN .
w(x) ≤ u(x) ≤ v(x),
(6)
Definition 2. Say that a function a(x) ∈ C(RN ), a(x) ≥ 0, has the property (H) if the linear problem −div |∇u| p−2 ∇u = a(
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