Journal ArticleDOI
On Schrödinger-Poisson Systems
TLDR
In this paper, the existence of bound states of the nonlinear Schrodinger-Poisson system has been studied in the context of critical point theory and perturbation methods.Abstract:
We discuss some recent results dealing with the existence of bound states of the nonlinear Schrodinger-Poisson system
$$\left\{ \begin{gathered} - \Delta u + V(x)u + \lambda K(x)\phi (x)u = |u|^{{p - 1}} u, \hfill \\ - \Delta \phi = K(x)u^{2}, \hfill \\ \end{gathered} \right.$$
as well as of the corresponding semiclassical limits. The proofs are based upon Critical Point theory and Perturbation Methods.read more
Citations
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Journal ArticleDOI
A guide to the Choquard equation
TL;DR: A survey of the existence and properties of solutions to the Choquard type equations can be found in this paper, where some variants and extensions of its variants can also be found.
Journal ArticleDOI
Positive solutions for some non-autonomous Schrödinger–Poisson systems
Giovanna Cerami,Giusi Vaira +1 more
TL;DR: In this article, the existence of positive solutions for the Schrodinger-Poisson system with nonnegative functions has been proved, but not requiring any symmetry property on them and satisfying suitable assumptions.
Journal ArticleDOI
Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system
TL;DR: In this paper, the existence of sign-changing solutions for the Schrodinger-Poisson system was investigated and invariant sets of descending flow invariants were used to prove that the system has infinitely many sign changing solutions.
Journal ArticleDOI
A guide to the Choquard equation
TL;DR: A survey of the existence and properties of solutions to the Choquard type equations can be found in this article, where some variants and extensions of its variants can also be found.
Journal ArticleDOI
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential
TL;DR: In this paper, the existence of nontrivial solution and concentration results are obtained via variational methods under suitable assumptions on V and K. In particular, the potential V is allowed to be sign-changing for the case p ∈ ( 4, 6 ).
References
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Journal ArticleDOI
A Simplification of the Hartree-Fock Method
TL;DR: In this article, the Hartree-Fock equations can be regarded as ordinary Schrodinger equations for the motion of electrons, each electron moving in a slightly different potential field, which is computed by electrostatics from all the charges of the system, positive and negative, corrected by the removal of an exchange charge, equal in magnitude to one electron, surrounding the electron whose motion is being investigated.
Journal ArticleDOI
On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN
TL;DR: In this paper, the authors derive a generic theorem for a wide class of functionals, having a mountain pass geometry, and show how to obtain, for a given functional, a special Palais-Smale sequence possessing extra properties that help to ensure its convergence.
Book
Perturbation Methods and Semilinear Elliptic Problems on R^n
TL;DR: In this paper, Pertubation in critical point theory is used to solve the Yamabe problem. But the problem in this paper is different from the ones in the present paper.
Journal ArticleDOI
Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I
TL;DR: In this paper, the existence of positive radial solutions concentrating on spheres to singularly perturbed elliptic problems was studied and necessary and sufficient conditions for concentration as well as the bifurcation of non-radial solutions were provided.