Abstract:
The aim of this article is twofold: firstly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem:
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−𝑎(∫Ω|∇𝑢|𝑁d𝑥)Δ𝑁𝑢=
𝑓(𝑥,𝑢)
|𝑥|𝑏
+𝜆ℎ(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)(Δ𝑁𝑢+𝑉(𝑥)|𝑢|𝑁−2𝑢)=
𝑔(𝑥,𝑢)
|𝑥|𝑏
+𝜆ℎ(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.