Abstract
We study the existence of ground states of a Schrödinger–Poisson–Slater type equation with pure power nonlinearity. By carrying out the constrained minimization on a special manifold, which is a combination of the Pohozaev manifold and Nehari manifold, we obtain the existence of ground state solutions of this system.
Highlights
Introduction and main resultIn [2] and [3], it is showed that the following minimization problem inf φ∈X 1,α\{0}Rn |∇φ|2dx p (n +α)−4n 2n −p (n −2)2(4+α−n) [L(φ)] 2(4+α−n) Rn |φ|p dx (1)is attained if and only if n = 4 + α and p ∈ p∈ 2(4+α) 2+α ∞ 2n n−2, n = 2; 3 ≤ n < 4 + α;
It is known that every solution to equation (4) is a critical point of the energy functional J : X 1,α → R, which is given by
By the implicit function theorem, it only need I (u) = 0 for any u ∈ M. We prove it by argument of contradiction
Summary
In [2] and [3], it is showed that the following minimization problem inf φ∈X 1,α\{0}. We are interested in studying the ground state solutions of (4) in X 1,α. It is known that every solution to equation (4) is a critical point of the energy functional J : X 1,α → R, which is given by. We point out that the usual Nehari manifold is not suitable because it is difficult to prove the boundedness of the minimizing sequences. A function u is called the Nehari–Pohozaev type ground state solution of (4), if u is a solution of the least energy problem min{J (u) : u ∈ M }. Problem (4) has a ground state solution of the Nehari–Pohozaev type
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
