التفاصيل البيبلوغرافية
| العنوان: |
On a class of elliptic equations with critical perturbations in the hyperbolic space. |
| المؤلفون: |
Ganguly, Debdip, Gupta, Diksha, Sreenadh, K. |
| المصدر: |
Asymptotic Analysis; 2024, Vol. 138 Issue 4, p225-253, 29p |
| مصطلحات موضوعية: |
ELLIPTIC equations, PERTURBATION theory, ENERGY levels (Quantum mechanics), NONLINEAR equations, HYPERBOLIC spaces, BLOWING up (Algebraic geometry), ENERGY consumption |
| مستخلص: |
We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a (x) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 (B N) , where B N denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞ , if N = 2 , λ < (N − 1) 2 4 , and 0 < a ∈ L ∞ (B N). We first prove the existence of a positive radially symmetric ground-state solution for a (x) ≡ 1. Next, we prove that for a (x) ⩾ 1 , there exists a ground-state solution for ε small. For proof, we employ "conformal change of metric" which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a (x) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space. [ABSTRACT FROM AUTHOR] |
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| قاعدة البيانات: |
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