Generalized Mountain Pass Theorem Research Articles

New Homoclinic Orbits for Hamiltonian Systems with Asymptotically Quadratic Growth at Infinity

In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity \begin{eqnarray*} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0. {eqnarray*} We introduce a new coercive condition and obtain a new embedding theorem. With this theorem, we show that above systems possess at least one nontrivial homoclinic orbits by Generalized Mountain Pass Theorem. By Variant Fountain Theorem, infinitely many homoclinic orbits are obtained for above problem with symmetric condition. Our asymptotically quadratic conditions are different from previous ones in the references.

Existence of periodic solutions for a class of damped vibration problems

In this paper, we are concerned with the existence of periodic solutions for a class of damped vibration problems. By introducing some new kinds of superquadratic and asymptotically quadratic conditions, and making use of the generalized mountain pass theorem in critical point theory, we propose a unified approach when the potential function F(t,x) exhibits either an asymptotically quadratic or a superquadratic behavior at infinity, and establish some sufficient conditions on periodic solutions, which extend and improve some recent results in the literature, even without damped vibration term.

Subharmonic solutions for a class of ordinary p-Laplacian systems ∗

In this paper, we study the existence of subharmonic solutions for ordinary p-Laplacian systems under a new growth condition. An existence theorem is obtained by using the generalized mountain pass theorem, which generalizes and improves some recent results in the literature.

Existence of nonconstant periodic solutions for a class of second-order systems with p(t)-Laplacian

In this paper, we investigate a class of second-order p(t)-Laplacian systems with local ‘superquadratic’ potential. By using the generalized mountain pass theorem, we obtain an existence result for nonconstant periodic solutions.

Nonconstant periodic solutions for a class of ordinary p-Laplacian systems

In this paper, we study the existence of periodic solutions for a class of ordinary p-Laplacian systems. Our technique is based on the generalized mountain pass theorem of Rabinowitz.

Erratum to: Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem

Erratum to: Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem

Periodic solution of second-order impulsive delay differential system via generalized mountain pass theorem

In this paper we use variational methods and generalized mountain pass theorem to investigate the existence of periodic solutions for some second-order delay differential systems with impulsive effects. To the authors’ knowledge, there is no paper about periodic solution of impulses delay differential systems via critical point theory. Our results are completely new.

SINGULAR POTENTIAL BIHARMONIC PROBLEM

We investigate the multiplicity of the solutions for a class of the system of the biharmonic equations with some singular potential nonlinearity. We obtain a theorem which shows the existence of the nontrivial weak solution for a class of the system of the biharmonic equations with singular potential nonlinearity and Dirichlet boundary condition. We obtain this result by using variational method and the generalized mountain pass theorem.

Fourth order elliptic boundary value problem with nonlinear term decaying at the origin

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.MSC:35J30, 35J40.

Existence of solutions for non-periodic superlinear Schrödinger equations without (AR) condition

The existence of solutions is obtained for a class of the non-periodic Schrödinger equation −Δu +V(x)u = f(x, u), x ∈ RN, by the generalized mountain pass theorem, where V is large at infinity and f is superlinear as |u| → ∞.

Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

A class of first‐order noncoercive discrete Hamiltonian systems are considered. Based on a generalized mountain pass theorem, some existence results of homoclinic orbits are obtained when the discrete Hamiltonian system is not periodical and need not satisfy the global Ambrosetti‐Rabinowitz condition.

Periodic and subharmonic solutions for a class of non-autonomous Hamiltonian systems

Some existence theorems are obtained for periodic and subharmonic solutions of a class of non-autonomous Hamiltonian systems. Our technical approach is based on a version of the Local Linking Theorem, and the Generalized Mountain Pass Theorem.

On homoclinic orbits for a class of noncoercive superquadratic Hamiltonian systems

In this paper, we prove the existence of homoclinic solutions for a class of noncoercive first order Hamiltonian systems J x ̇ − M ( t ) x + u ∗ G ′ ( t , u ( x ) ) = 0 , by the minimax methods in critical point theory, specially, a Generalized Mountain Pass Theorem, when u is a linear operator with adjoint u ∗ and G ( t , y ) satisfies the superquadratic condition G ( t , x ) | y | 2 ⟶ − + ∞ as | y | ⟶ ∞ , uniformly in t , and need not satisfy the global Ambrosetti–Rabinowitz condition.

Existence and multiplicity of homoclinic solutions for a class of damped vibration problems

The main purpose of this paper is to study the following damped vibration problems (1.1) { − u ̈ ( t ) − B u ̇ ( t ) + A ( t ) u ( t ) = ∇ F ( t , u ( t ) ) a.e. t ∈ R u ( t ) → 0 , u ̇ ( t ) → 0 as | t | → ∞ where A = [ a i , j ( t ) ] ∈ C ( R , R N 2 ) is an N × N symmetric matrix-valued function, B = [ b i j ] is an antisymmetry N × N constant matrix, F ∈ C 1 ( R × R N , R ) and ∇ F ( t , u ) : = ∇ u F ( t , u ) . By a symmetric mountain pass theorem and a generalized mountain pass theorem, an existence result and a multiplicity result of homoclinic solutions of (1.1) are obtained.

Periodic and subharmonic solutions for a class of superquadratic second order Hamiltonian systems

Some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic second order Hamiltonian systems by the minimax methods in critical point theory, especially a local linking theorem due to Luan and Mao, and the Generalized Mountain Pass Theorem due to Rabinowitz.

Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential

Some existence theorems are obtained for periodic solutions of a class of non-autonomous second order Hamiltonian systems with local superquadratic potential, by making use of the generalized mountain pass theorem.

Periodic and subharmonic solutions for a class of superquadratic Hamiltonian systems

Some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic Hamiltonian systems by the minimax methods in critical point theory, specially, a local link theorem due to Li and Willem, and the Generalized Mountain Pass Theorem respectively.

SOME REMARKS ON CRITICAL POINT THEORY FOR LOCALLY LIPSCHITZ FUNCTIONS Work performed under the auspices of G.N.A.M.P.A. of I.N.D.A.M. and partially supported by M.I.U.R. of Italy, 2001.

In this paper, a dual version of the Mountain Pass Theorem and the Generalized Mountain Pass Theorem are extended to functions that are locally Lipschitz only. An application involving elliptic hemivariational inequalities is next examined.

On the Existence of Nontrivial Solutions to Some Elliptic Variational Inequalities

Abstract The paper is concerned with the existence of nontrivial solutions of the obstacle problem: where K = {υ ∈ H1 0(Ω) : υ ≤ ψ a.e. on Ω}. By using a generalized mountain pass theorem for inequalities, we prove, subject to some restrictions on the obstacle ψ, the existence of nontrivial solutions of the above inequality.

Nonlinear Elliptic Eigenvalue Problems with Discontinuities

In this paper we study the existence of solution for two different eigenvalue problems. The first is nonlinear and the second is semilinear. Our approach is based on results from the nonsmooth critical point theory. In the first theorem we prove the existence of at least two nontrivial solutions when λ is in a half-axis. In the second theorem (based on a nonsmooth variant of the generalized mountain pass theorem), we prove the existence of at least one nontrivial solution for every λ∈R.