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This paper is concerned with the existence of solutions to the Kirchhoff-type equation − a + b ∫ R 3 | ∇ u | 2 Δ u + (V + λ) u = | u | p − 2 u + μ | u | q − 2 u i n R 3 under the normalized constraint ∫ R 3 u 2 = ρ 2 , where a, b, ρ > 0, 14/3 < q < p ⩽ 6, μ > 0 is a constant, and λ ∈ R appears as a Lagrange multiplier. Under an explicit assumption on V, we can prove the existence of positive ground state solutions to the above equation. A new concentration compactness type result is established to recover compactness in the Sobolev critical case. [ABSTRACT FROM AUTHOR]

In this work, we investigate the (2, p)-Laplacian equation −Δu − Δpu = f(x, u) in Ω with the boundary condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in R N , p > 2, and the nonlinearity f has extension property at both the zero and infinity points. We observe that the above equation admits at least two positive solutions, owing to the mountain pass theorem and Ekeland's variational principle. [ABSTRACT FROM AUTHOR]