Normalized solutions of mass supercritical Schrödinger–Poisson equation with potential

In this paper we prove the existence of normalized solutions (lambda, u), subset of (0, infinity) x H-1(R-3) to the following Schrodinger-Poisson equation { -Delta u + V(x)u + )u + (|x|(-1) * u(2))u = |u|(p-2)u in R-3, u > 0, integral(3)(R) u(2)dx = a(2), where a > 0 is fixed, p is an element of ( 10/3 , 6) is a given exponent and the potential V satisfies some suitable conditions. Since the L-2(R-3)-norm of u is fixed, ) appears as a Lagrange multiplier. For V (x) >= 0, our solutions are obtained by using a mountain-pass argument on bounded domains and a limit process introduced by Bartsch et al (Commun Partial Differ Equ 46:1729-1756, 2021). For V (x) <= 0, we directly construct an entire mountain-pass solution with positive energy.