Non-trivial solutions of fractional Schrödinger-Poisson systems with sum of periodic and vanishing potentials
Texas State University, Department of Mathematics
We consider the fractional Schrödinger-Poisson system (-Δ)αu + V(x)u + K(x)Φ(x)u = ƒ(x, u) - Γ(x)|u|q-2u in ℝ3, (-Δ)βΦ = K(x)u2 in ℝ3, where α, β ∈ (0, 1], 4α + 2β > 3, 4 ≤ q < 2*α, K(x), Γ(x) and ƒ(x, u) are periodic in x, V is coercive or V = Vper + Vloc is a sum of a periodic potential Vper and a localized potential Vloc. If ƒ has the subcritical growth, but higher than Γ(x)|u|q-2u, we establish the existence and nonexistence of ground state solutions are dependent on the sign of Vloc. Moreover, we prove that such a problem admits infinitely many pairs of geometrically distinct solutions provided that V is periodic and ƒ is odd in u. Finally, we investigate the existence of ground state solutions in the case of coercive potential V.
Fractional Schrödinger-Poisson system, Coercive potential, Periodic and localized potential, Nehari manifold, Variational method
Yu, M., & Chen, H. (2019). Non-trivial solutions of fractional Schrödinger-Poisson systems with sum of periodic and vanishing potentials. <i>Electronic Journal of Differential Equations, 2019</i>(102), pp. 1-16.