Landesman-Lazer Type Condition Research Articles

Dynamic bifurcation of nonautonomous evolution equations under Landesman–Lazer condition with cohomology methods

In this article we study the dynamic bifurcation of nonautonomous evolution equations by using cohomology methods. First, we construct a homotopy equivalence relation between the nonautonomous system and a product flow. Then, we slightly extend some continuation theorems on bifurcations for autonomous equations, and prove some new cohomology consequences on the reduced singular groups. Based on this homotopy equivalence relation and these conclusions, we establish some typical results on the dynamic bifurcation from infinity of the abstract nonautonomous evolution equation. Finally, we consider the parabolic equation ut−Δu=λu+f(x,u)+g(x,t) associated with the Dirichlet boundary condition, where f(x,u) satisfies the appropriate Landesman–Lazer type condition. Some new results on the dynamical behaviors of this equation near resonance of the equation are derived.

Bifurcation from infinity and multiplicity results for an elliptic system from biology

This article is concerned with the bifurcation from infinity of the following elliptic system arising from biology − κ Δ u = λ u + f ( x , u ) − v , − Δ v = u − v , in a bounded domain Ω ⊂ R N . We regard this problem as a stationary problem of some reaction-diffusion system. By using a method of a pure dynamical nature, we will establish some multiplicity results on bifurcations from infinity for this system under an appropriate Landesman-Lazer type condition.

Landesman-Lazer type conditions for scalar one-sided superlinear nonlinearities with Neumann boundary conditions

In this paper, we provide an existence result for a scalar differential equation ruled by a nonlinearity near resonance presenting a one-sided superlinear condition. The result is achieved by the introduction of a suitable Landesman-Lazer type condition. The main theorem can be extended also to the case of equations with a singularity. The proof is given by using a shooting method technique.

Solvability for a fractional $ p $-Laplacian equation in a bounded domain

<abstract><p>In this paper we use the topological degree and the fountain theorem to study the existence of weak solutions for a fractional $ p $-Laplacian equation in a bounded domain. For the nonlinearity $ f $, we consider two situations: (1) the non-resonance case where $ f $ is $ (p-1) $-asymptotically linear at infinity; (2) the resonance case where $ f $ satisfies the Landesman-Lazer type condition.</p></abstract>

Bifurcation from infinity of the Schrödinger equation via invariant manifolds

This paper is concerned with the bifurcation from infinity of the nonlinear Schrödinger equation −Δu+V(x)u=λu+f(x,u),x∈RN. We treat this problem in the framework of dynamical systems by considering the corresponding parabolic equation on unbounded domains. Firstly, we establish a global invariant manifold for the parabolic equation on RN. Then, we restrict the parabolic equation to this invariant manifold, which generates a system of finite dimension. Finally, we use the Conley index theory and the shape theory of attractors to establish some new results on bifurcations from infinity and multiplicity of solutions of the Schrödinger equation under an appropriate Landesman–Lazer type condition.

Landesman-Lazer type (p, q)-equations with Neumann condition

We consider a Neumann problem driven by the (p, q)-Laplacian under the Landesman-Lazer type condition. Using the classical saddle point theorem and other classical results of the calculus of variations, we show that the problem has at least one nontrivial weak solution.

On global dynamics of reaction–diffusion systems at resonance

In this paper we use the homotopy invariants methods to study the global dynamics of the reaction–diffusion systems that are at resonance at infinity. Considering degrees of the resonance for the nonlinear perturbation we establish Landesman–Lazer type conditions and use them to express the Rybakowski–Conley index of the invariant set consisting of all bounded solutions. Obtained results are applied to study the existence of solutions connecting stationary points for the system of nonlinear heat equations.

Morse homology for asymptotically linear Dirac equations on compact manifolds

We give constructions and calculations of Morse (co)homologies for asymptotically linear Dirac equations on compact manifolds which are non-resonance at infinity, or resonance at infinity and satisfy Landesman-Lazer type conditions. For the non-resonance case, we show that the homology depends only on the homotopy class of the linear part at infinity, while for the resonance case, the (co)homology is isomorphic with the Morse (co)homology of the restriction of the perturbation term to the degenerate space at infinity. From these, (co)homologies are explicitly calculated. Applications to the existence of solutions of asymptotically linear Dirac equations is also given.

Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

Existence of solutions for a resonant problem under Landesman-Lazer type conditions involving more general elliptic operators in divergence form

Some bifurcation results for fractional Laplacian problems

We consider the following nonlocal problem (−Δ)su=λu+f(λ,x,u)inΩ;u=0inRN∖Ω,where (−Δ)s is the fractional Laplacian operator with fixed 0<s<1, Ω⊂RN(N>2s) is a bounded domain with C1,1 boundary, CN,s>0 is a constant and λ∈R is a bifurcation parameter. Here f:R×Ω×R→R is a Carathéodory function that is sublinear at infinity. We use bifurcation theory to establish the existence of continua of the solution set bifurcating from infinity at the principal eigenvalue of (−Δ)s and discuss the nodal properties of solutions on these continua. We establish the multiplicity of solutions near the resonance and the existence of solution in the resonant case. As corollaries, we obtain anti-maximum principle and solvability for the resonant case satisfying the so called Landesman–Lazer type condition.

Existence and Multiplicity of Solutions for p-Laplacian Neumann Problems

In this paper, existence theorems are proved for p-Laplacian Neumann problems under the Landesman–Lazer type condition. Our results are derived from a classical saddle point theorem and the least action principle respectively. Furthermore, multiplicity of solutions is established by applying a known multiple critical points result due to H. Brezis and L. Nirenberg. The above-mentioned conclusions are based on variational methods.

On Fractional p-Laplacian Equations at Resonance

This article shows the existence of weak solutions of a resonant problem for a fractional p-Laplacian equation in a bounded domain in \({\mathbb {R}}^N\). Our arguments are based on the Minimum principle, saddle point theorem and rely on a generalization of the Landesman–Lazer-type condition.

Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions

We consider reaction–diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub- or supercritical. We obtain both Landesman–Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

Periodic Impact Motions at Resonance of a Particle Bouncing on Spheres and Cylinders

Abstract We investigate the existence of periodic trajectories of a particle, subject to a central force, which can hit a sphere or a cylinder. We will also provide a Landesman–Lazer-type condition in the case of a nonlinearity satisfying a double resonance condition. Afterwards, we will show how such a result can be adapted to obtain a new result for the impact oscillator at double resonance.

Resonant Sturm–Liouville Boundary Value Problems for Differential Systems in the Plane

We study the Sturm-Liouville boundary value problem associated with the planar differential system Jz'=\nabla V(z) + R(t, z) , where V(z) is positive and positively 2 -homogeneous and R(t, z) is bounded. Assuming Landesman-Lazer type conditions, we obtain the existence of a solution in the resonant case. The proofs are performed via a shooting argument. Some applications to boundary value problems associated with scalar second order asymmetric equations are discussed.

Retraction Note: Landesman-Lazer type condition for second-order differential equations at resonance with impulsive effects

Retraction Note: Landesman-Lazer type condition for second-order differential equations at resonance with impulsive effects

A Landesman–Lazer type result for periodic parabolic problems on [formula omitted] at resonance

We are concerned with T-periodic solutions of nonautonomous parabolic problem of the form ut=Δu+V(x)u+f(t,x,u), t>0, x∈RN, with V∈L∞(RN)+Lp(RN), where 2<p<∞ if N=1,2 and N≤p<∞ for N≥3 and T-periodic continuous perturbation f:RN×R→R. The so-called resonant case is considered, i.e. when N:=Ker(Δ+V)≠{0} and f is bounded. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer topological degree of the time averaging of the mapping fˆ:N→N, fˆ being the restriction of f to N. By use of the formula and continuation techniques we show that Landesman–Lazer type conditions imply the existence of T-periodic solutions.

Existence of weak solutions for a class of (p,q)-Laplacian systems on resonance

The existences of weak solutions for a class of ( p , q ) -Laplacian systems on resonance are obtained under the certain Landesman–Lazer-type conditions by using G-link Theorem due to Drábek and Robinson (1999). Our results give a positive answer to an open problem of Zographopoulos (2004).

On existence of weak solutions for a p-Laplacian system at resonance

This article shows the existence of weak solutions of a resonance problem for uniformly p-Laplacian system in a bounded domain in \(R^N\). Our arguments are based on the Saddle Point Theorem (P.H.Rabinowitz) and rely on a generalization of the Landesman–Lazer type condition.