We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth, where ε is a small positive parameter and a, b > 0 are constants, f ∈ C ¹ (ℝ ⁺ ,ℝ) is subcritical, V : ℝ ³ → ℝ is a locally Hölder continuous function. We first prove that for ε ₀ > 0 sufficiently small, the above problem has a weak solution u _ε with exponential decay at infinity. Moreover, u _ε concentrates around a local minimum point of V in Λ as ε → 0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(z) attains its minimum.

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation . Let . For any integer k ≥ 1, we prove existence of standing wave solutions with u > 0 having k local maximum points and concentrating near a given local maximum point of Γ when ε is small. The potentials V and K are allowed to be either vanishing or unbounded at infinity. Existence of solutions concentrating near k distinct non-degenerate critical points of Γ has been proved under the same assumptions on V and K as well.

We consider the following nonlinear Schrödinger system with mixed couplings in $\mathbb {R}^3$: \begin{equation*} -\Delta u_i + \lambda _i u_i=\mu _i u_i^3+\sum \limits _{j=1,j\neq i}^N\beta _{ij}u_j^2u_i,\ \ \ i=1,\cdots ,N, \end{equation*} where $\lambda _i, \mu _i>0, \beta _{ij}=\beta _{ji} (i,j=1,\cdots ,N, i\neq j)$. The system appears in modeling of Bose-Einstein condensates theory. While most existing works in the literature are concerned with purely attractive or purely repulsive couplings (i.e., all $\beta _{ij}$ have the same signs), we examine the effect of mixed nonlinear couplings on the solution structure and obtain vector solutions with some of the components synchronized between them while being segregated with the rest of the components simultaneously.