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In this paper, we consider the following nonlinear Schrödinger equations with mixed nonlinearities:where N ≥ 3, µ > 0, λ ∈ R and 2 < q < 2 * . We prove in this paper (1) Existence of solutions of mountain-pass type for N = 3 and 2 < q < 2 + 4 N .(2)Existence and nonexistence of ground states for 2 + 4 N ≤ q < 2 * with µ > 0 large.(3)Precisely asymptotic behaviors of ground states and mountain-pass solutions as µ → 0 and µ goes to its upper bound. Our studies answer some questions proposed by Soave in [49, Remarks 1.1, 1.2 and 8.1].

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Consider the following Kirchhoff type problemwhereN −2 and a, b, λ, µ are positive parameters. By introducing some new ideas and using the well-known results of the problem (P) in the cases of a = µ = 1 and b = 0, we obtain some special kinds of solutions to (P) for all N ≥ 3 with precise expressions on the parameters a, b, λ, µ, which reveals some new phenomenons of the solutions to the problem (P). It is also worth to point out that it seems to be the first time that the solutions of (P) can be expressed precisely on the parameters a, b, λ, µ, and our results in dimension four also give a partial answer to Neimen's open problems

In this paper, we study the following concave–convex elliptic problems: [Formula: see text] where N ≥ 3, 1 < q < 2 < p < 2* = 2N/(N - 2), λ > 0 and μ < 0 are two parameters. By using several variational methods and a perturbation argument, we obtain three positive solutions to this problem under the predefined conditions of fλ(x) and gμ(x), which simultaneously extends the result of [T. Hsu, Multiple positive solutions for a class of concave–convex semilinear elliptic equations in unbounded domains with sign-changing weights, Bound. Value Probl. 2010 (2010), Article ID 856932, 18pp.; T. Wu, Multiple positive solutions for a class of concave–convex elliptic problems in ℝN involving sign-changing weight, J. Funct. Anal. 258 (2010) 99–131]. We also study the concentration behavior of these three solutions both for λ → 0 and μ → -∞.