Low Perturbations for a Class of Nonuniformly Elliptic Problems
- PDF / 369,220 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 224 Views
Low Perturbations for a Class of Nonuniformly Elliptic Problems Anouar Bahrouni
and Duˇsan D. Repovˇs
Abstract. In this paper, we introduce and study a new functional which was motivated by the work of Bahrouni et al. (Nonlinearity 31:1518– 1534, 2018) on the Caffarelli–Kohn–Nirenberg inequality with variable exponent. We also study the eigenvalue problem for equations involving this new functional. Mathematics Subject Classification. Primary 35J60; Secondary 35J91, 58E30. Keywords. Caffarelli–Kohn–Nirenberg inequality, eigenvalue problem, critical point theorem, generalized Lebesgue–Sobolev space, Luxemburg norm.
1. Introduction The Caffarelli–Kohn–Nirenberg inequality plays an important role in studying various problems of mathematical physics, spectral theory, analysis of linear and nonlinear PDEs, harmonic analysis, and stochastic analysis. We refer to [2,4,7,8] for relevant applications of the Caffarelli–Kohn–Nirenberg inequality. Let Ω ⊂ RN (N ≥ 2) be a bounded domain with smooth boundary. The following Caffarelli–Kohn–Nirenberg inequality [5] establishes that given p ∈ (1, N ) and real numbers a, b, and q, such that: −∞ < a
0, and s ∈ (1, +∞), such that: (1) |a(x)| = 0, for every x ∈ Ω \ {x0 }; (2) |a(x)| ≥ |x − x0 |s , for every x ∈ B(x0 , r); (P) p : Ω → R is a function of class C 1 and 2 < p(x) < N for every x ∈ Ω. Theorem 1.1. (Bahrouni et al. [3]) Suppose that hypotheses (A) and (P ) are satisfied. Let Ω ⊂ RN (N ≥ 2) be a bounded domain with smooth boundary. Then, there exists a positive constant β, such that: |a(x)|p(x) |u(x)|p(x) dx ≤ β |a(x)|p(x)−1 ||∇a(x)||u(x)|p(x) dx Ω Ω +β |a(x)|p(x) |∇u(x)|p(x) dx Ω p(x) p(x)+1 |a(x)| |∇p(x)||u(x)| dx + Ω +β |a(x)|p(x)−1 |∇p(x)||u(x)|p(x)−1 dx. Ω
for every u ∈ Cc1 (Ω). Motivated by [3], we introduce and study in the present paper a new functional T : E1 → R via the Caffarelli–Kohn–Nirenberg inequality, in the framework of variable exponents. More precisely, we study the eigenvalue problem in which functional T is present. Our main result is Theorem 4.2 and we prove it in Sect. 5.
2. Function Spaces with Variable Exponent We recall some necessary properties of variable exponent spaces. We refer to [10,12,13,15–17] and the references therein. Consider the set: C+ (Ω) = {p ∈ C(Ω) | p(x) > 1 for all x ∈ Ω}. For any p ∈ C+ (Ω), let p+ = s
Data Loading...
