Nontrivial Homoclinic Solutions Research Articles

The Existence and Multiplicity of Homoclinic Solutions for a Fractional Discrete p−Laplacian Equation

In this study, we investigate the existence and multiplicity of solutions for a fractional discrete p−Laplacian equation on Z. Under suitable hypotheses on the potential function V and the nonlinearity f, with the aid of Ekeland’s variational principle, via mountain pass lemma, we obtain that this equation exists at least two nonnegative and nontrivial homoclinic solutions when the real parameter λ>0 is large enough.

Homoclinic solutions of discrete nonlinear Schrödinger equations with unbounded potentials

In this paper, the existence and multiplicity of homoclinic solutions for discrete nonlinear Schrödinger equations −Δuk+vkuk−ωuk=fk(uk)are considered. With asymptotical nonlinearity and unbounded potential V={vk:k∈Z} satisfying lim∣k∣→∞vk=∞, we achieve the existence of nontrivial homoclinic solution via the saddle point reduction method and Conley index theory. Furthermore, we can prove there are at least two nontrivial homoclinic solutions with extra assumptions about the nonlinearity fk.

Existence of two homoclinic solutions for a nonperiodic difference equation with a perturbation

<abstract> In the present paper, with the combination of a compactness lemma and variational techniques, we establish a sufficient condition to guarantee the existence of two nontrivial homoclinic solutions for a nonperiodic fourth-order difference equation with a perturbation. Our result generalizes and improves some known results. </abstract>

The Nehari manifold method for discrete fractional p-Laplacian equations

The aim of this paper is to investigate the multiplicity of homoclinic solutions for a discrete fractional difference equation. First, we give a variational framework to a discrete fractional p-Laplacian equation. Then two nontrivial and nonnegative homoclinic solutions are obtained by using the Nehari manifold method.

Homoclinic solutions for a class of nonlinear fourth order [formula omitted]-Laplacian differential equations

The aim of this paper is the study of existence of homoclinic solutions for a class of nonlinear differential equations involving p-Laplacian. Under suitable growth conditions we prove that the considered problem has at least one nontrivial homoclinic solution. The proof is based on the mountain-pass theorem with Cerami’s condition, Brezis–Lieb lemma and variational method.

Perturbed Schrödinger lattice systems: existence of homoclinic solutions

AbstractWe study a class of Schrödinger lattice systems with sublinear nonlinearities and perturbed terms. We get an interesting result that the systems do not have nontrivial homoclinic solutions if the perturbed terms are removed, but the systems have ground state homoclinic solutions if the perturbed terms are added. Besides, we also study the continuity of the homoclinic solutions in the perturbation terms at zero. To the best of our knowledge, there is no published result focusing on the perturbed Schrödinger lattice systems.

New homoclinic solutions for a class of second-order Hamiltonian systems with a mixed condition

In this paper, we introduce a new mixed condition to obtain a new compact embedding theorem. Under this theorem, we study the existence and multiplicity of nontrivial homoclinic solutions for a class of second-order Hamiltonian systems with variable separated type nonlinear terms.

Two homoclinic solutions of a nonperiodic fourth-order impulsive differential equation

In this paper, we are concerned with a kind of nonperiodic fourth-order impulsive differential equation. By applying the mountain pass theorem, we shall establish several new criteria for the existence of at least two nontrivial homoclinic solutions of the addressed equation. Some recent results in the literature are generalized and improved. In addition, a numerical example is provided to show the applicability and effectiveness of our main results.

Multiple homoclinic solutions for a class of nonhomogeneous Hamiltonian systems

By introducing a new superquadratic condition, we obtain the existence of two nontrivial homoclinic solutions for a class of perturbed second order Hamiltonian systems which are obtained by the mountain pass theorem and Ekeland’s variational principle.

Existence of solutions for nonlinear p-Laplacian difference equations

The aim of this paper is the study of existence of solutions for nonlinear $2n^{\mathrm{th}}$-order difference equations involving $p$-Laplacian. In the first part, the existence of a nontrivial homoclinic solution for a discrete $p$-Laplacian problem is proved. The proof is based on the mountain-pass theorem of Brezis and Nirenberg. Then, we study the existence of multiple solutions for a discrete $p$-Laplacian boundary value problem. In this case the proof is based on the three critical points theorem of Averna and Bonanno.

Local super-quadratic conditions on homoclinic solutions for a second-order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second-order Hamiltonian system u ̈ − L ( t ) u + ∇ W ( t , u ) = 0 , where t ∈ R , u ∈ R N , L : R → R N × N and W : R × R N → R . Applying the Mountain Pass Theorem, we prove the existence of nontrivial homoclinic solutions under new weaker conditions. In particular, we use a local super-quadratic condition lim | x | → ∞ W ( t , x ) | x | 2 = ∞ , uniformly in t ∈ ( a , b ) for some − ∞ < a < b < + ∞ , instead of the common one lim | x | → ∞ W ( t , x ) | x | 2 = ∞ , uniformly in t ∈ R , which is essential to show the existence of nontrivial homoclinic solutions for the above system in all existing literature.

New Results for Second Order Discrete Hamiltonian Systems

In this paper, we deal with the second order discrete Hamiltonian system $\Delta[p(n) \Delta u(n-1)] - L(n) u(n) + \nabla W(n,u(n)) = 0$, where $L\colon \mathbb{Z} \to \mathbb{R}^{N \times N}$ is positive definite for sufficiently large $|n| \in \mathbb{Z}$ and $W(n,x)$ is indefinite sign. By using critical point theory, we establish some new criteria to guarantee that the above system has infinitely many nontrivial homoclinic solutions under the assumption that $W(n,x)$ is asymptotically quadratic and supquadratic, respectively. Our results generalize and improve some existing results in the literature.

Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities

In this paper, we study the existence of homoclinic solutions to the following second-order Hamiltonian systems \begin{eqnarray} \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\quad \forall t\in R, \end{eqnarray} where $L(t)$ is a symmetric and positive definite matrix for all $t\in R$. The nonlinear potential $W$ is a combination of superlinear and sublinear terms. By different conditions on the superlinear and sublinear terms, we obtain existence and nonuniqueness of nontrivial homoclinic solutions to above systems.

Existence of time homoclinic solutions for a class of discrete wave equations

In this paper, a class of nonperiodic discrete wave equations with Dirichlet boundary conditions are obtained by using the center-difference method. It is a strongly indefinite discrete Hamiltonian system. By using a variant and generalized weak linking theorem, the existence of the nontrivial time homoclinic solutions for the system will be obtained. The obtained main results here allow the classical Ambrosetti-Rabinowitz superlinear condition to be replaced by a general superquadratic condition. Such a method cannot be used for the corresponding continuous wave equations, however, it is valid for some general discrete Hamiltonian systems. Similarly, the existence of homoclinic periodic solutions can also be considered.

Homoclinic solutions for a class of prescribed mean curvature Liénard equations

In this paper, the problem of a homoclinic solution is studied for the prescribed mean curvature Lienard equation $(\frac{u'(t)}{\sqrt{1+(u'(t))^{2}}})'+f(u(t))u'(t)+g(u(t))=p(t)$ , where $f\in C(R,R)$ , $g\in C^{1}(R,R)$ , and $p\in C(R,R)$ . Under some conditions, the author obtains the result that the equation has at least one nontrivial homoclinic solution for $p(t)\not\equiv0$ , and the equation has no nontrivial homoclinic solution for $p(t)\equiv0$ . The arguments are based upon Mawhin’s continuation theorem.

Nontrivial homoclinic solutions for prescribed mean curvature Rayleigh equations

In this paper, some sufficient conditions on the existence of \(2k\pi\)-periodic solutions for a kind of prescribed mean curvature Rayleigh equations are given. Then the existence of nontrivial homoclinic solutions for prescribed mean curvature Rayleigh equations is obtained.

Multiple homoclinic orbits for second order discrete Hamiltonian systems without symmetric condition

In this paper, we investigate the second order self-adjoint discrete Hamiltonian system $\Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\lambda a(n)\nabla G(u(n))+\mu b(n)\nabla F(u(n))=0$ , where $p,L:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N}$ are both positive definite for all $n\in\mathbb{Z}$ , and no symmetric condition on G and F is needed. We establish two new criteria to guarantee that the above system has at least two nontrivial homoclinic solutions or infinitely many homoclinic solutions via critical point theory.

Existence of homoclinic solutions for a fourth order differential equation with a parameter

In this paper, we study the existence of homoclinic solutions for a class of fourth order differential equations. By using variational methods, the existence and the non-existence of nontrivial homoclinic solutions are obtained, depending on a parameter.

Homoclinic solutions for a nonperiodic fourth order differential equations without coercive conditions

In this paper we investigate the existence of homoclinic solutions for the following fourth order nonautonomous differential equationsu(4)+wu″+a(x)u=f(x,u),(FDE)wherew is a constant, a∈C(R,R) and f∈C(R×R,R). The novelty of this paper is that, when (FDE) is nonperiodic, i.e., a and f are nonperiodic in x and assuming that a does not fulfil the coercive conditions and f satisfies some more general (AR) condition, we establish one new criterion to guarantee that (FDE) has at least one nontrivial homoclinic solution via using the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.

New potential condition on homoclinic orbits for a class of discrete Hamiltonian systems

In the present paper, we establish an existence criterion to guarantee that the second-order self-adjoint discrete Hamiltonian system △[p(n)△u(n−1)]−L(n)u(n)+∇W(n,u(n))=0 has a nontrivial homoclinic solution, which does not need periodicity and coercivity conditions on L(n).MSC:39A11, 58E05, 70H05.