Multiple normalized solutions for a Sobolev critical Schrodinger-Poisson-Slater equation

We look for solutions to the Schrodinger-Poisson-Slater equation -Delta u+lambda u - gamma(vertical bar x vertical bar(-1) * vertical bar u vertical bar(2))u - a vertical bar u vertical bar(p-2) u=0 in R-3, (0.1) which satisfy parallel to u parallel to(L2(R3))(2) = c for some prescribed c > 0. Here u is an element of H-1 (R-3), gamma is an element of R, a is an element of R and p is an element of (10/3,6]. when gamma > 0 and a >0, both in the Sobolev subcritical case p is an element of(10/3,6) and in the Sobolev critical case p = 6, we show that there exists a c(1) > 0 such that, for any c is an element of(0, c(1)), (0.1) admits two solutions u(c)(+) and u(c)(-) which can be characterized respectively as a local minima and as a mountain pass critical point of the associated Energy functional restricted to the norm constraint. In the case gamma > 0 and a < 0, we show that, for any p is an element of(10/3, 6] and any c > 0, (0.1) admits a solution which is a global minimizer. Finally, in the case gamma < 0, a > 0 and p = 6 we show that (0.1) does not admit positive solutions. (C) 2021 Elsevier Inc. All rights reserved.