Variant Fountain Theorem Research Articles

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.

Non-periodic damped vibration systems with sublinear terms at infinity: Infinitely many homoclinic orbits

In this paper, we study a class of non-periodic damped vibration systems with subquadratic terms. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence of infinitely many homoclinic orbits for this class of non-periodic damped vibration systems with subquadratic terms at infinity.

On multiple solutions of a semilinear Schrödinger equation with periodic potential

This paper is concerned with the semilinear Schrödinger equation (S)−Δu+V(x)u=f(x,u),u∈H1(RN), where V and f are periodic in the x-variables, f is a superlinear and subcritical nonlinearity, and 0 lies in a spectral gap of −Δu+V. It is shown that, if f is odd in u then (S) has infinitely many large energy solutions. The proof relies on a generalized variant fountain theorem for strongly indefinite functionals, established in this paper.

Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems

In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem.

Nonperiodic Damped Vibration Systems with Asymptotically Quadratic Terms at Infinity: Infinitely Many Homoclinic Orbits

We study a class of nonperiodic damped vibration systems with asymptotically quadratic terms at infinity. We obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by Zou. To the best of our knowledge, there is no result published concerning the existence (or multiplicity) of nontrivial homoclinic orbits for this class of non-periodic damped vibration systems with asymptotically quadratic terms at infinity.

Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations

In this paper we study the existence of infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations. The proof is based on the variant fountain theorem established by Zou. Recent results from the literature are extended.

Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems

By a variant version of Fountain Theorem due to Zou [W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358], the existence of infinitely many solutions is obtained for a class of biharmonic equations where the nonlinearity involves a combination of superlinear and asymptotically linear terms.

The Existence and Multiplicity of Solutions for Second-order Impulsive Differential Equations on the Half-line

In this paper, we consider the existence and multiplicity of solutions for a class of nonlinear impulsive problem on the half-line. By using a variational method and a variant fountain theorem, we obtain some new criteria to guarantee that the impulsive problem has at least one solution and infinitely many solutions respectively. We extend and improve some recent results.

Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems

Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems

Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems

In this paper, we study the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. By using the variant fountain theorem, we obtain a new criterion for guaranteeing that second-order Hamiltonian systems has infinitely many homoclinic solutions. Recent results from the literature are generalized and significantly improved. An example is also given in this paper to illustrate our main results.

Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems

In this paper, we study the existence of infinitely many classical solutions for a class of second-order impulsive differential equations. By using two new fountain theorems, we deal with two cases: that when the nonlinearity is superlinear and that when it is asymptotically linear at infinity. Some recent results are extended and improved.

Existence of infinitely many large solutions for the nonlinear Schrödinger–Maxwell equations

In this paper, we study the nonlinear stationary Schrödinger–Maxwell equations (∗) { − Δ u + V ( x ) u + ϕ u = f ( x , u ) , in R 3 , − Δ ϕ = u 2 , in R 3 . Using the variant fountain theorem introduced by Zou [W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358], under certain assumptions on V and f , we get infinitely many large solutions for (∗).

Multiplicity of solutions for a class of elliptic boundary value problems

In this paper we study the existence of infinitely many solutions for a class of elliptic boundary value problems. The existence results include both superlinear case and an asymptotically linear case. Our arguments are based on a recently given new Fountain Theorem due to Zou [W. Zou, Variant Fountain Theorems and their applications, Manuscripta Math. 104 (2001) 343–358].

Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials

We study the existence of infinitely many homoclinic orbits for some second-order Hamiltonian systems: u ̈ − L ( t ) u ( t ) + ∇ F ( t , u ( t ) ) = 0 , ∀ t ∈ R , by the variant fountain Theorem, where F ( t , u ) satisfies the super-quadratic condition F ( t , u ) / | u | 2 → ∞ as | u | → ∞ uniformly in t, and need not satisfy the global Ambrosetti–Rabinowitz condition.

On the existence of a solution for elliptic system related to the Maxwell–Schrödinger equations

In this paper we study the existence and the multiplicity of standing wave solutions for the nonlinear Schrödinger equation coupled with the Maxwell equations. It seems difficult to obtain a boundedness of a Palais–Smale sequence for the functional associated with this system. We overcome this difficulty by Jeanjean’s result [L. Jeanjean, On the existence of bounded Palais–Smale sequence and application to a Landesman–Lazer type problem set on R N , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787–809], which generalizes Struwe’s argument [M. Struwe, Variational Methods, 2nd ed., Springer, 1996; M. Struwe, The existence of a surface of constant mean curvature with free boundaries, Acta Math. 160 (1988) 19–64] and Zou’s result [W. Zou, Variant fountain theorem and their applications, Manuscripta Math. 104 (2001) 343–358].

Variant fountain theorems and their applications

In this paper we establish some variant fountain theorems without (P.S.)-type assumption. The abstract results will be used to study the symmetric nonlinear Schrodinger equations and Dirichlet boundary value problems. Under no Ambrosetti–Rabinowitz's superquadraticity condition, we obtain infinitely many large energy and small negative energy solutions respectively.