Twin Positive Solutions for Schrödinger-Kirchhoff-Type Problem with Singularity and Critical Exponents. (2nd May 2018) - Han, Wei; Zhao, Yangyang
Abstract : We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent- a + b ∫ Ω ∇ u 2 d x Δ u + u = Q ( x ) u 5 + μ x α - 2 u + f ( x ) ( λ / u γ ) inΩ, u = 0 on∂ Ω, wherea, b > 0 are constants, Ω ⊂ R 3 is a smooth bounded domain, 0 < α < 1, λ > 0 is a real parameter, γ ∈ ( 0, 1 ) is a constant, and0 < μ < a μ 1 (μ 1 is the first eigenvalue of- Δ u = μ x α - 2 u, under Dirichlet boundary condition). Under appropriate assumptions onQ andf, we obtain two positive solutions via the variational and perturbation methods.