Existence of Solutions for the Fractional ( p, q )-Laplacian Problems Involving a Critical Sobolev Exponent
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
EXISTENCE OF SOLUTIONS FOR THE FRACTIONAL (p, q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT∗
~~)
Fanfan CHEN (
)
Yang YANG (
†
School of Science, Jiangnan University, Wuxi 214122, China E-mail : [email protected]; [email protected] Abstract In this article, we study the following fractional (p, q)-Laplacian equations involving the critical Sobolev exponent: (−∆)s1 u + (−∆)s2 u = µ|u|q−2 u + λ|u|p−2 u + |u|p∗s1 −2 u, in Ω, p q (Pµ,λ ) u = 0, in RN \ Ω,
where Ω ⊂ RN is a smooth and bounded domain, λ, µ > 0, 0 < s2 < s1 < 1, 1 < q < p < sN1 . We establish the existence of a non-negative nontrivial weak solution to (Pµ,λ ) by using the Mountain Pass Theorem. The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers. Key words
fractional (p, q)-Laplacian; non-negative solutions; critical Sobolev exponents
2010 MR Subject Classification
1
35B33; 35R11
Introduction and Main Results
In this article, we consider the following fractional (p, q)-Laplacian equation involving critical Sobolev exponent: (−∆)s1 u + (−∆)s2 u = µ|u|q−2 u + λ|u|p−2 u + |u|p∗s1 −2 u, in Ω, p q (Pµ,λ ) u = 0, in RN \ Ω,
where Ω ⊂ RN is a smooth, bounded domain, λ, µ > 0, 0 < s2 < s1 < 1, 1 < q < p < p s p∗s1 = N N −s1 p , and the nonlocal operator (−∆)a (a ≥ 1) is defined as follows: Z |u(x) − u(y)|a−2 (u(x) − u(y)) (−∆)sa u(x) = 2 lim dy, x ∈ RN . ε→0 B (x) |x − y|N +sa ε
N s1 ,
The equation in (Pµ,λ ) arises while studying the stationary solutions of the general reactionaldiffusion equation ut = div(a(u)∇u) + r(x, u), (1.1) ∗ Received
September 14, 2019; revised July 26, 2020. This work was supported by National Natural Science Foundation of China (11501252 and 11571176). † Corresponding author: Yang YANG.
No.6
F.F. Chen & Y. Yang: THE FRACTIONAL (p, q)-LAPLACIAN PROBLEMS
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where a(u) = |∇u|p−2 + |∇u|q−2 . Problem (1.1) has wide real world applications in fields such as biophysics [15], plasma physics [19], reaction-diffusion equations [3, 12], and models of elementary particles [5, 13]. In these contexts, u represents a concentration, div(a(u)∇u) is the diffusion with a diffusion coefficient a(u), and the reaction term r(x, u) relates to source and loss processes. Typically, in chemical and biological applications, the reaction term r(x, u) has a polynomial form with respect to the concentration u. These wide applications have prompted many researchers to study stationary solutions of (1.1) as −div(a(u)∇u) = r(x, u), which can be found in [6] on a bounded domain Ω ⊂ RN . On the other hand, the eigenvalue problem for a Laplacian type equation with p = 2 was investigated in [12]. For the critical (p, q)-Laplacian problem −∆ u − ∆ u = µ|u|r−2 u + |u|p∗ −2 u, in Ω, p q u = 0, on ∂Ω,
where Ω ⊂ RN is a smooth, bounded domain, µ is a positive parameter, 1 < q < p < N , and p p∗ = NN−p is the critical Sobolev
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