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Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan ZengChun Lei Tang — 2016

Annales Polonici Mathematici

We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ - [ a + b ( Ω | u | ² d x ) m ] Δ u = f ( x , u ) + | u | 2 * - 2 u in Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω N (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Jia-Feng LiaoJiu LiuPeng ZhangChun-Lei Tang — 2016

Annales Polonici Mathematici

We study the following singular elliptic equation with critical exponent ⎧ - Δ u = Q ( x ) u 2 * - 1 + λ u - γ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where Ω N (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu LeiChang-Mu ChuHong-Min SuoChun-Lei Tang — 2015

Annales Polonici Mathematici

In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧ - ( a + b Ω | u | ² d x ) Δ u = u + λ u q - 1 / | x | β in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

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