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In this paper, we prove the multiplicity and concentration behavior of complex-valued solutions for the following Kirchhoff–Schrödinger equation with magnetic field - (a ε 2 + b ε [ u ] A / ε 2) Δ A / ε u + V (x) u = f (| u | 2) u , x ∈ R 2 , where ε > 0 is a small parameter, the nonlinearity f is involved in critical exponential growth in the sense of Trudinger–Moser inequality and both V : R 2 → R and A : R 2 → R 2 are continuous potential and magnetic potential, respectively. Imposing a local constraint of potential V(x) first introduced from del Pino and Felmer, we get the multiplicity of solutions by way of the relationship between the number of the solutions and the topology of the set with V attaining the minimum. Our strategy of main proof is based on the variational methods combined with the penalization technique, the Trudinger–Moser inequality and Ljusternik–Schnirelmann theory, and our result is still new even without magnetic effect. [ABSTRACT FROM AUTHOR]

We devote this paper to the multiplicity and concentration behavior of complex-valued solutions for the following Schrödinger–Poisson equation with magnetic Laplacian (ε i ∇ - A (x) ) 2 u + V (x) u - ϕ (x) | u | 3 u = f (| u | 2 ) u + | u | 4 u in R 3 , - ε 2 Δ ϕ = | u | 5 in R 3 , where ε > 0 is a small parameter, the nonlinearity f ∈ C 1 (R , R) , both V : R 3 → R and A : R 3 → R 3 are continuous potential and magnetic potential, respectively. Imposing a global condition on potential V(x) first introduced from Rabinowitz, to get the multiplicity of solutions we investigate the relationship between the number of solutions and the topology of a set with V attaining the minimum based on the variational methods and Lusternik–Schnirelmann theory. Moreover, we prove that the problem under consideration admits the solutions with exponential decay at infinity and concentrating around global minimum of V(x), for ε > 0 small. We would like to remark that a key ingredient is how to apply a delicate analysis to overcome the difficulty due to the presence of double critical growths and magnetic field. [ABSTRACT FROM AUTHOR]