Existence and multiplicity of solutions to N-Kirchhoff equations with critical exponential growth and a perturbation term

Abstract

The aim of this article is twofold: firstly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: ⎧ ⎪ ⎨ ⎪ ⎩ −𝑎⁡(∫Ω|∇𝑢|𝑁d𝑥)⁢Δ𝑁⁢𝑢= 𝑓⁡(𝑥,𝑢) |𝑥|𝑏 +𝜆⁢ℎ⁡(𝑥)in Ω,𝑢=0on ∂Ω, where Ω is a smooth bounded domain in ℝ𝑁⁢(𝑁≥ 2) and 0≤𝑏<𝑁. Secondly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: −𝑎⁡(∫ℝ𝑁|∇𝑢|𝑁+𝑉⁡(𝑥)⁢|𝑢|𝑁d𝑥)⁢(Δ𝑁⁢𝑢+𝑉⁡(𝑥)⁢|𝑢|𝑁−2⁢𝑢)= 𝑔⁡(𝑥,𝑢) |𝑥|𝑏 +𝜆⁢ℎ⁡(𝑥)in ℝ𝑁, where 𝑁≥ 2 and 0≤𝑏<𝑁. We assume that f and g have critical exponential growth at infinity. To establish our existence results, we use the mountain pass theorem, Ekeland variational principle and Moser–Trudinger inequality.