Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms

Based on new information concerning strongly indefinite functionals without Palais-Smale conditions, we study existence and multiplicity of solutions of the Schrödinger equation

     

Sharp regularity for functionals with (p,q) growth

We prove regularity theorems for minimizers of integral functionals of the Calculus of Variations

     

Multiplicity for strongly indefinite semilinear elliptic system

In this paper we study a multiplicity result for a strongly indefinite semilinear elliptic system

     

On finding sign-changing solutions

Some parameter-depending linking theorems are established, which allow to produce a bounded and sign-changing Palais-Smale sequence. For even functionals, a parameter-depending fountain theorem is obtained which provides infinitely many bounded and sign-changing Palais-Smale sequences. A variant mountain pass theorem is built in cones which yields bounded, positive and negative Palais-Smale sequences. The usual Palais-Smale type compactness condition and its variants are completely not necessary for these theories. More exact locations of the critical sequences can be determined. The abstract results are applied to the Schrödinger equation with (or without) critical Sobolev exponents:

     

An infinite-dimensional linking theorem and applications

In this paper, we establish a new infinite-dimensional linking theorem without (PS)-type assumptions. The new theorem needs a weaker linking geometry and produces bounded (PS) sequences. The abstract result will be applied to the study of the existence of solutions of the strongly indefinite partial differential systems. For the first application, we consider the system

     

On a functional inequality connected with quadratic functionals

In the present paper we investigate some functional inequalities which are closely connected with quadratic functionals. In particular, we are interested in inequalities of the type

     

Second-order energies on thin structures: variational theory and non-local effects

For a given positive measure μ on , we consider integral functionals of the kind

     

Oscillation of self-adjoint linear matrix Hamiltonian systems

By means of monotone functionals defined on suitable matrix spaces and new methods, oscillation criteria for self-adjoint linear Hamiltonian matrix system of the form

     

Wong's comparison theorem for second order linear dynamic equations on time scales

We obtain Wong-type comparison theorems for second order linear dynamic equations on a time scale. The results obtained extend and are motivated by Wong's comparison theorems. As a particular application of our results, we show that the difference equation

     

Infinitely many solutions to perturbed elliptic equations

A new version of perturbation theory is developed which produces infinitely many sign-changing critical points for uneven functionals. The abstract result is applied to the following elliptic equations with a Hardy potential and a perturbation from symmetry:

     

On the validity of the Euler-Lagrange equation in a nonlinear case

We prove the validity of the Euler-Lagrange equation for the class of functionals of the type

     

On the ?ojasiewicz-Simon gradient inequality

We prove a general version of the ?ojasiewicz-Simon inequality, and we show how to apply the abstract result to study energy functionals E of the form

     

Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II

This paper is a continuation of the paper [T.Y. Lee, Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral, J. Math. Anal. Appl. 298 (2004) 677-692], in which we proved several Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. Let f be Henstock-Kurzweil integrable on a compact interval . For a given compact interval , set

     

Critical groups at infinity for p-Laplacian equations with indefinite nonlinearities

In this paper, by a kind of decomposition lemma and Künneth formula we study the critical groups at infinity for the associated functional of the following p-Laplacian equation with indefinite nonlinearities

     

Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem

We study Brezis-Nirenberg type theorems for the equation

     

Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals

In this paper, we prove Krasnoselskii and Mann's type convergence theorems for nonexpansive semigroups without using Bochner integral and without assuming the strict convexity of Banach spaces. One of our main results is the following: let C be a compact convex subset of a Banach space E and let be a one-parameter strongly continuous semigroup of nonexpansive mappings on C. Let {tn} be a sequence in [0,∞) satisfying

     

Sign-changing saddle point

In this paper, the Saddle-point theorems are generalized to a new version by showing that there exists a “sign-changing” saddle point besides zero. The abstract result is applied to the semilinear elliptic boundary value problem