The Nehari Manifold for Indefinite Kirchhoff Problem With Caffarelli–Kohn–Nirenberg Type Critical Growth

Document Type

Article

Publication Date

9-1-2021

Publication Title

Topological Methods in Nonlinear Analysis

Publisher

Juliusz Schauder Centre for Nonlinear Studies

Publisher Location

Toruń

Volume

58

Issue

1

First page number:

105

Last page number:

134

Abstract

In this paper we study the following class of nonlocal problem involving Caffarelli–Kohn–Nirenberg type critical growth L(u) − λh(x)|x|−2(1+a) u = µf(x)|u|q−2 u + |x|−pb |u|p−2 u in ℝN, where h(x) ≥ 0, f(x) is a continuous function which may change sign, λ, µ are positive real parameters and 1 < q < 2 < 4 < p = 2N/[N +2(b −a) −2], 0 ≤ a < b < a + 1 < N/2, N ≥ 3. Here (∫ ) L(u) = −M |x|−2a |∇u|2 dx div(|x|−2a ∇u) ℝN and the function M: ℝ+0→ℝ+0 is exactly the Kirchhoff model, given by M(t) = α + βt, α, β > 0. The above problem has a double lack of compact-ness, firstly because of the non-compactness of Caffarelli–Kohn–Nirenberg embedding and secondly due to the non-compactness of the inclusion map ∫ u ↦→ h(x)|x|−2(a+1) |u|2 dx, ℝN as the domain of the problem in consideration is unbounded. Deriving these crucial compactness results combined with constrained minimization argument based on Nehari manifold technique, we prove the existence of at least two positive solutions for suitable choices of parameters λ and µ.

Keywords

Caffarelli–Kohn–Nirenberg; Critical growth; Kirchhoff; Nehari manifold

Disciplines

Applied Mathematics | Numerical Analysis and Computation

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