Landesman-Lazer Type Condition Research Articles

Resonance problems for the [formula omitted]-Laplacian with a nonlinear boundary condition

Consider resonance problems at an arbitrary eigenvalue of the elliptic equations ▵ p u = | u | p - 2 u - g 1 ( u ) + h 1 ( x ) , in Ω , | ∇ u | p - 2 ∂ u ∂ ν = λ | u | p - 2 u + g 2 ( u ) + h 2 ( x ) , on ∂ Ω . Some existence theorems are obtained under the Landesman–Lazer type conditions by using variational method.

Periodic solutions for second order differential equations

A class of second order ordinary differential equations is considered. It is shown that the equation considered possesses unique 2 π -periodic solution under Landesman–Lazer type conditions.

Multiple solutions for resonant semilinear elliptic problems in [formula omitted

We prove the existence of multiple nontrivial solutions for the semilinear elliptic problem − Δ u = h ( λ u + g ( u ) ) in R N , u ∈ D 1 , 2 , where h ∈ L 1 ∩ L α for α > N / 2 , N ⩾ 3 , g is a C 1 ( R , R ) function that has at most linear growth at infinity, g ( 0 ) = 0 , and λ is an eigenvalue of the corresponding linear problem − Δ u = λ h u in R N , u ∈ D 1 , 2 . Existence of multiple solutions, for certain values of g ′ ( 0 ) , is obtained by imposing a generalized Landesman–Lazer type condition. We use the saddle point theorem of Ambrosetti and Rabinowitz and the mountain pass theorem, as well as a Morse-index result of Ambrosetti [A. Ambrosetti, Differential Equations with Multiple Solutions and Nonlinear Functional Analysis, Equadiff 82, Lecture Notes in Math., vol. 1017, Springer-Verlag, Berlin, 1983] and a Leray–Schauder index theorem for mountain pass type critical points due to Hofer [H. Hofer, A note on the Topological Degree at a critical Point of Mountain Pass Type, Proc. Amer. Math. Soc. 90 (1984) 309–315]. The results of this paper are based upon multiplicity results for resonant problems on bounded domains in [E. Landesman, S. Robinson, A. Rumbos, Multiple solutions of semilinear elliptic problems at resonance, Nonlinear Anal. 24 (1995) 1049–1059] and [S. Robinson, Multiple solutions for semilinear elliptic boundary value problems at resonance, Electron. J. Differential Equations 1995 (1995) 1–14], and complement a previous existence result by the authors in [G. López Garza, A. Rumbos, Resonance and strong resonance for semilinear elliptic equations in R N , Electron. J. Differential Equations 2003 (2003) 1–22] for resonant problems in R N in which g was assumed to be bounded.

An nn-dimensional pendulum-like equation via topological methods

We study the elliptic boundary value problem Δu+g(u)=p(x) with nonlinear periodic-type boundary conditions. We prove the existence of a solution under a Landesman–Lazer-type condition. Moreover, using the method of upper and lower solutions we study a problem which concerns the range of the semilinear operator Su=Δu+g(u).

An [formula omitted]-dimensional pendulum-like equation via topological methods

We study the elliptic boundary value problem Δ u + g ( u ) = p ( x ) with nonlinear periodic-type boundary conditions. We prove the existence of a solution under a Landesman–Lazer-type condition. Moreover, using the method of upper and lower solutions we study a problem which concerns the range of the semilinear operator Su = Δ u + g ( u ) .

Non-linear second-order periodic systems with non-smooth potential

In this paper we study second order non-linear periodic systems driven by the ordinary vectorp-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by thep-Laplacian. In the last section of the paper we examine the scalar non-linear and semilinear problem. Our approach uses a generalized Landesman-Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.

P-Laplacian Systems on Resonance

We assume certain (adapted to the case of a system) Landesman–Lazer-type conditions, in order to prove existence results for a p-Laplacian system on resonance.

Bounded solutions of second order semilinear evolution equations and applications to the telegraph equation

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Differential inclusions at resonance

The existence of periodic solutions are studied for certain differential inclusions at resonance. Landesman–Lazer type conditions are derived. Applications are given to discontinuous differential equations.

On a Third Order Nonlinear Boundary Value Problem at Resonance

In this paper we discuss the existence of solutions to third order boundary value problems of the form x‴ + x′ + g(x, x′) = p(t),x′(0) = x′(π) = x(η) = 0, where η lies in [0, π]. We assume that g is continuous and one-sided bounded (e.g., g ≥ 0) and p is continuous. Solutions are shown to exist if p satisfies a Landesman-Lazer type condition involving g. Several examples are given to illustrate how these conditions depend upon the parameter η.

Landesman-Lazer condition for nonlinear problems with jumping nonlinearities

This paper was motivated by the result of Iannacci and Nkashama ( J. Differential Equations 69 (1987) , 289–309) concerning the existence of at least one solution for the differential equation x″( t) + m 2 x( t) + g( t, x( t)) = e( t) with periodic boundary data x(0) − x(2 π) = x′(0) − x′(2 π) = 0, where m ⩾ 0 is an integer, e is integrable, and g satisfies Carathéodory's conditions. We intend to prove that the Landesman-Lazer type condition is sufficient for solvability of this periodic problem under rather more general assumptions on the growth of the nonlinear term g. Particularly, our hypotheses cover the nonlinearities which may “jump over” the eigenvalues different from m 2 while the assumptions of Iannacci and Nkashama allow g only to asymptotically “touch” the neighbour eigenvalues. Our approach may be used also for two-point boundary value problems. The proofs are based on the Leray-Schauder degree theory and on the shooting method for ordinary differential equations.

Landesman-Lazer type condition and nonlinearities with linear growth

Landesman-Lazer type condition and nonlinearities with linear growth

Unbounded nonlinear perturbations of linear elliptic problems at resonance

435 where lj is an eigenvalue of L, a/& is the derivative in the direction outward normal to the boundary &2 of 0, and f~ H-“(Q). Recently, Shapiro [lo] established an existence result for the problem (1.1) in the case that g has unbounded nonlinearities and satisfies a Landesman-Lazer type condition. Shapiro’s result, however, does not cover the case that g has linear growths. Our first result is an extension of Shapiro’s result to the case that g has a linear growth. We next consider the case g does not satisfies any Landesman-Lazer type condition. That is, we study the case lim inf, _ oc g(x, t) = lim supI _ up g(x, t), a.e. on Q. Several authors considered this case with g having bounded nonlinearities (cf. [9]). Fucik and Krebec [5] and Hess [6] gave existence results for (1.1) in case that j?‘~ L”(Q) and g is a continuous function satisfying g(x, t) = g(t), g(-t)=g(r), and g+ =lim,+ fm g(t) = 0. We extend these results for (1.1) to the case g has unbounded nonlinearities and f~ L2. Our method employed here is different from those in [S, 6, or lo]. 2.

Periodic solutions of equations with nonlinearities involving the derivatives

has no root of the form 2kni/T with k a nonzero integer. When this choice for L is made, equation (1) becomes a generalized Liknard equation. Such equations have been studied by Cesari and Kannan [l] and Mawhin [4] when written in system form and by FuGk [2] as a scalar equation. Fu8k [2] has investigated the generalized Liinard equation when g is bounded and strictly monotone and f is bounded. Landesman-Lazer type conditions are then given on p to ensure the existence of a T-periodic solution. In the present paper, by the use of alternative methods, we will prove existence theorems where g need not be bounded nor strictly monotone but satisfies a generalization of monotonicity (se (N2)). Nonlinearities such as in equation (2) are not considered in [14]. By obtaining bounds for x’ we are able to give existence results for equation (2) where the conditions imposed are similar to those used for equation (1).