Fuyi Li

Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent

The existence of positive solutions to Schrodinger–Poisson type systems in ℝ3 with critically growing nonlocal term is proved by using variational method which does not require usual compactness conditions. A key ingredient of the proof is a new Brezis–Lieb type convergence result.

Existence and uniqueness results for Kirchhoff–Schrödinger–Poisson system with general singularity

Abstract In this paper, under the general singular assumptions on f, we discuss the existence and uniqueness of solution to a class of Kirchhoff–Schrödinger–Poisson system. For the existence of solution, we assume that f is nonincreasing on and satisfies general singularity at zero, and sublinear growth at infinity. Furthermore, we also obtain the unique result assuming that f satisfies one side Lipschitz condition on . The variational method is employed to discuss the existence and uniqueness of solution to this system.

Existence of ground state solutions to a generalized quasilinear Schrödinger-Maxwell system

In this paper, a class of generalized quasilinear Schrodinger-Maxwell systems is considered. Via the mountain pass theorem, we conclude the existence of positive ground state solutions when the potential may vanish at infinity and the nonlinear term has a quasicritical growth. During this process, we use the Coulomb energy studied by Ruiz [Arch. Ration. Mech. Anal. 198(1), 349–368 (2010)] and establish a convergency theorem to overcome the lack of compactness caused by the potential which may vanish at infinity.

Existence of multiple positive solutions to nonhomogeneous Schrödinger-Poisson system

In this paper, we consider the existence of multiple solutions to the following nonhomogeneous generalized Schrodinger-Poisson system - Δ u + Ku + q ? f ( u ) = g ( u ) + h ( x ) , in R 3 , - Δ ? = 2 qF ( u ) , in R 3 , where q ? 0 is a parameter, 0 ? h ( x ) = h ( | x | ) ? L 2 ( R 3 ) , and g is asymptotically linear or superliner at infinity. We show that there exists q 0 0 such that the system has at least two positive radial solutions for q ? 0 , q 0 ) .

Positive solutions of Kirchhoff-type non-local elliptic equation: a bifurcation approach

Positive solutions of a Kirchhoff-type nonlinear elliptic equation with a non-local integral term on a bounded domain in ℝ N , N ⩾ 1, are studied by using bifurcation theory. The parameter regions of existence, non-existence and uniqueness of positive solutions are characterized by the eigenvalues of a linear eigenvalue problem and a nonlinear eigenvalue problem. Local and global bifurcation diagrams of positive solutions for various parameter regions are obtained.

A new sufficient condition for blow-up of solutions to a class of parabolic equations

In this paper, a new sufficient condition under which the solutions to a classical parabolic equation blow up is established. During this process, through constructing a new functional which falls between the energy and Nehari functionals, we also get a larger blow-up set.

Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term

In this paper, we investigate an initial boundary value problem to a class of pseudo-parabolic partial differential equations with Newtonian nonlocal term. First, the local existence and uniqueness of a weak solution is established. In virtue of the energy functional and the related Nehari manifold, we also describe the exponent decay behavior and the blow up phenomenon of weak solutions with different kinds of initial data. Our second conclusion states that some solutions starting in a potential well exist globally, whereas solutions with suitable initial data outside the potential well must blow up. Furthermore, the instability of a ground state equilibrium solution is studied.

Existence of solutions to a class of Schrödinger–Poisson systems with indefinite nonlinearity

In this paper, we investigate a class of Schrödinger–Poisson systems with indefinite nonlinearity which is a combination of a linear term with parameter and a superlinear term with parameter . Here, the Poisson equation is the form , where . A concentration compactness lemma is established to overcome the lack of compactness. In order to insure the Nehari manifold , the parameters and must have some restriction that compares with and the variation range of depends on l. By seeking the local minimizer of the energy functional on the Nehari manifold, we obtain the existence of solution for the system.