Critical Schrödinger-Bopp-Podolsky systems in R 3: existence of solutions and limit behaviour - Damian, Heydy Melchora Santos (2025)
In this work, we first study the following critical Schrödinger-Bopp-Podolsky system, in R^3: -epsilon^2 Delta u + V(x)u + Q(x)phi u = h(x,u) + K(x)|u|^4u, -Delta phi + a^2Delta^2 phi = 4 pi Q(x)u^2, where the unknowns are u, phi: R^3 --> R, and epsilon,a > 0 are arbitrary parameters. The functions V, K, Q satisfy suitable assumptions as well as the nonlinearity h which is subcritical. For any fixed a> 0, we show existence of small solutions in the semiclassical limit, namely whenever epsilon tends to zero. We give also estimates of the norm of this solutions in terms of epsilon. Moreover, we show also that fixed epsilon suitably small, when a tends to zero, the solutions found strongly converge to solutions of the Schrödinger-Poisson system. Additionally, we study the following Schrödinger-Bopp-Podolsky system with critical and sublinear terms in R^3: -Delta u + u + R(x)phi u = |u|^4u + lambda W(x)|u|^(p-1)u, -Delta phi + a^2 Delta^2 phi = 4pi R(x)u^2. Here u,phi: R^3 --> R are the unknowns, R and W are given functions satisfying mild assumptions, a >= 0, lambda > 0 are parameters and p in (0,1). We first show existence of infinitely many solutions at negative energy level, including the ground state, when the parameter lambda is small. Then we give general results concerning the structure of the set of solutions. We show also the behaviour of the solutions as the parameters a, lambda tend to zero. In particular the ground states solutions tends to a ground state solution of the Schrödinger-Poisson system as a tends to zero.