Solutions with prescribed mass for the Sobolev critical Schrödinger–Poisson system with p-Laplacian

Abstract

In this paper, we are concerned with the existence and properties of ground states for the quasilinear Schrödinger–Poisson system with combined critical nonlinearities −Δpu+γϕ|u|p−2u=λ|u|p−2u+μ|u|q−2u+|u|p∗−2uin ℝ3,−Δϕ=|u|pin ℝ3, having prescribed mass ∫ℝ3|u|pdx=ap, in the Sobolev critical case. Here, [Formula: see text] and [Formula: see text], [Formula: see text] are parameters, [Formula: see text] is the Sobolev critical exponent, and [Formula: see text] is an undetermined parameter, appeared as a Lagrange multiplier. By using Jeanjean’s theory, Pohozaev manifold analysis and the Brezis–Nirenberg technique to overcome the lack of compactness, we prove several existence results in the [Formula: see text]-subcritical, [Formula: see text]-critical and [Formula: see text]-supercritical perturbation [Formula: see text], under different assumptions imposed on the parameters [Formula: see text] and the mass a, respectively. To the best of our knowledge, this work seems to be the first contribution regarding the existence of normalized solutions of the Sobolev critical Schrödinger–Poisson problem with p-Laplacian, perturbed with a subcritical term in the whole space [Formula: see text].

Keywords

• p-Laplacian Schrödinger–Poisson systems
• normalized solutions
• critical Sobolev exponent
• Pohozaev manifold