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Many problems arising in science and engineering call for the solving of the Euler-Lagrange equations of functionals. Thus, solving the Euler-Lagrange equations is tantamount to finding critical points of the corresponding functional. An idea that has been very successful is to find appropriate sets that separate the functional. This method is called linking. Two sets A , B are said to separate a functional G if the supremum of G on A is less than or equal to the infimum of G on B. Two sets of the space are said to link if they produce a critical sequence whenever they separate a functional. If the critical sequence has a convergent sub-sequence, then that produces a critical point. Finding sets that separate a functional is quite easy, but determining whether or not the sets link is quite another story. It appears that the only way we can check to see if two sets link, is to require that one of them be contained in a finite-dimensional subspace. The reason is that in order to verify the definition, we need to invoke the Brouwer fixed point theorem. Our aim is to find a counterpart of linking that holds true when both sets are infinite dimensional. We adjust our definitions to accommodate infinite dimensions. These definitions reduced to the usual definitions when one set is finite dimensional. In order to prove the corresponding theorems, we make adjustments to the topology of the space and introduce infinite dimensional splitting. This allows us to use a form of compactness on infinite dimensional subspaces which does not exist under the usual topology. We lose the Brouwer index, but we are able to replace it with the Leray-Schauder index. We carry out the details in Sections 5 and 6 . In Section 7 we solve a system of equations which require infinite dimensional splitting.

We study the existence of periodic solutions for a second order non-autonomous dynamical system containing variable kinetic energy terms. Subquadratic problems and superquadratic problems are both considered.

We study the nonlinear Schrodinger equation in $$\mathbb {R}^n$$ without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in $$[0, \infty )$$ being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.

The Sandwich Pair theorems have presented very efficient ways to determine the existence of critical points or critical sequences for nonlinear differentiable functionals. In this paper, under rather weak hypotheses new relationships are established between sign-changing critical points and Sandwich Pairs or Linking Sandwich Pairs. The abstract results are demonstrated by applications on semi-linear elliptic equations. © 2013 Elsevier Ltd. All rights reserved.

We find nontrivial and ground state solutions for the nonlinear Schrodinger equation under conditions weaker than those previously assumed.

We show how monotonicity methods combined with infinite-dimensional sandwich pairs can be used to solve very general systems of equations that are not semibounded.

We use Morse theoretic arguments to obtain nontrivial solutions of semilinear elliptic boundary value problems where the nonlinearity interacts with the spectrum, without assuming that the asymptotic limits at zero and infinity exist.

We show how hypotheses for many problems can be significantly reduced if we employ the monotonicity method. We apply it to problems for the semilinear wave equation, where infinite dimensional methods are needed.

We apply some of the methods of sandwich pairs to semilinear equations and systems.

We solve boundary value problems for p -Laplacian systems using sandwich pairs.

For many equations arising in practice, the solutions are critical points of functionals. In previous papers we have shown that there are pairs of subsets, called sandwich pairs, that can produce critical points even though they do not separate the functional. All that is required is that the functional be bounded from above on one of the sets and bounded from below on the other, with no relationship needed between the bounds. This provides a distinct advantage in applications. The present paper discusses the situation in which one cannot find sandwich pairs for which the functional is bounded below on one set and bounded above on the other. We develop a method which can deal with such situations and apply it to problems in partial differential equations.

We use flows and the cohomological index to adapt the method of sandwich pairs to better suit quasilinear elliptic boundary value problems.

The variational approach to solving many nonlinear problems eventually leads to the search for critical points of related functionals. In the case of semibounded functionals, one can look for extrema. Otherwise, one is forced to use other methods. There are several approaches to such methods. In this paper we examine some successful approaches and exhibit their effectiveness.

© 2015 . We consider periodic solutions for superlinear second order non-autonomous dynamical systems including both kinetic and potential terms. We study the existence of nontrivial and ground state solutions.

We study the existence of periodic solutions for a second-order non-autonomous dynamical system. We give three sets of hypotheses which guarantee the existence of non-constant solutions. We were able to weaken the hypotheses considerably from those used previously for such systems. We employ a saddle point theorem using linking methods.

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: - Δ u - μ u | x | 2 = f ( x , u ) + p ( x , u ) in Ω , u = 0 on ∂ Ω and - Δ u = | u | q - 2 | x | s u + p ( x , u ) in Ω , u = 0 on ∂ Ω , where 0 s 2 , Ω is a smooth bounded domain of R n , and p ( x , u ) is not odd in u . Infinitely many sign-changing solutions are obtained.

From the beginning of the calculus of variations over 300 years ago, scientists were interested in finding extrema of expressions (now called functionals) involving unknown functions. It was discovered that functions producing extrema are solutions of differential equations (called Euler equations). In one dimentional problems these were ordinary differential equations, and the easiest procedure was to solve the corresponding Euler equation and then check if the solution produces the desired maximum or minimum. However, as researchers began to study higher dimensional problems, it became obvious that it was more difficult to solve the Euler equations than obtain an extremum of the functional. Moreover, many partial differential equations and systems that were being studied by various scientists happened to be the Euler equations of functionals. In this case, any critical point of the corresponding functional will provide a solution of the given equation. This led to the search for critical points of functionals. In this paper we describe several methods of finding critical points. We present applications.

In this paper, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A general liking type theorem is given with the location of the critical point produced specified in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities, and may or may not be resonant with respect to Fucik spectrum.

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 Δ u=u in Ω , Δ v=v in Ω , ∂u/∂η=H v (x,u,v) on ∂ Ω , ∂v/∂η=H u (x,u,v) on ∂ Ω , where Ω is a bounded domain in R N with smooth boundary, ∂ ∂η is the outer normal derivative, H : ∂ Ω × R × R → R is a positive C 1 -function. One nontrivial solution is obtained. The second application, we will solve the eigenvalue problem of the system A v=−f(x,v,w), B w=βg(x,v,w), β>0, where A , B are self-adjoint operators on L 2 ( Ω ) , Ω ⊂ R N is not necessarily bounded; f,g are Caratheodory functions on Ω × R 2 . We get infinitely many solutions. We deal with asymptotically linear cases for both systems.

In this paper, we establish a new double linking theorem: if two sets A and B link each other, then, under suitable conditions, we get two bounded (PS)-sequences which produce two distinct critical points. The critical points either have different critical values or belong to linking sets A and B separately. The abstract result will be used to solve semilinear eigenvalue problems and problems with jumping nonlinearities which may or may not resonant with respect to Fucik spectrum.

We obtain nontrivial solutions of semilinear elliptic boundary value problems with jumping nonlinearities that have resonance with respect to the Fucik spectrum both at zero and at infinity.

We study the existence of periodic solutions for a second order non-autonomous dynamical system containing variable kinetic energy terms. Our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other. We study sublinear problems and the existence of non-constant solutions. © 2014 Elsevier Inc.

We obtain nontrivial solutions for semilinear elliptic boundary value problems having asymptotic limits both at zero and at infinity.

We obtain nontrivial solutions of semilinear boundary value problems with jumping nonlinearities that interact with the Fucik spectrum.

We obtain nontrivial solutions for semilinear elliptic boundary value problems having resonance both at zero and at infinity, when the nonlinear term has asymptotic limits.

The Fucik spectrum for a semilinear problem with asymptotic linearities has been shown to consist, at least locally, of curves emanating from a sequence of points in the plane. Regions between curves emanating from different points (referred to as type (I) regions in this paper) have a different nature than those between curves emanating from the same point (referred to as type (II) regions). Problems for which asymptotic limits fall in regions of type (I) have been solved by several authors, but not those for which the limits fall in a type (II) region. In the present paper we solve problems in which the asymptotic limits fall in type (II) regions.

We prove a saddle point theorem for a functional using linking subsets of splitting subspaces. The values of the functional on one of the linking subsets do not have to be bounded away from those values on the other subset. Moreover, it does not matter which subspace is finite dimensional. Applications are given to boundary value problems for elliptic partial differential equations.

1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 Mathematisches Institut, University Giessen, Arndtstr. 2, 35392 Giessen, Germany 3 Graduate School of Environment and Information Science, Yokohama National University, Tokiwadai, Yokohama 240-8501, Japan 4 Mathematics Department, University of California, Irvine, CA 92697-3875, USA 5 Department of Mathematics and Statistics, Utah State University, Logan, UT 84322-4205, USA

nor do we assume more smoothness than mere continuity of g(x, t) in t. In particular, no assumption concerning the modulus of continuity of g(x, t) in I is made. Castro and Lazer [l] studied (0.1) under the assumptions that A, < A’ 5 g(x, tyt I A” < A/+, and ag(x, t)/& is increasing in It I. Ambrosetti and Mancini [2] consider the situation when g(x, t) = &r + J-(x, r) with f(x, f) uniformly bounded. Thews [3) assumed (1J + h,(x, f))/f 5 g(x, t)/t 5 (&+it hz(x, t))/t where the hi are bounded and satisfy

The concept of linking was developed to produce Palais-Smale (PS) sequencesG(uk)→a,G'(uk)→0forC1functionalsGthat separate linking sets. These sequences produce critical points if they have convergent subsequences (i.e., ifGsatisfies the PS condition). In the past, we have shown that PS sequences can be obtained even when linking does not exist. We now show that such situations produce more useful sequences. They not only produce PS sequences, but also Cerami sequences satisfyingG(uk)→a,(1+||uk||)G'(uk)→ 0as well. A Cerami sequence can produce a critical point even when a PS sequence does not. In this situation, it is no longer necessary to show thatGsatisfies the PS condition, but only that it satisfies the easier Cerami condition (i.e., that Cerami sequences have convergent subsequences). We provide examples and applications. We also give generalizations to situations when the separating criterion is violated.

Under suitable conditions we are able to solve the semilinear wave equation in any dimension. We are also able to compute the essential spectrum of the linear wave operator for the rotationally invariant periodic case.

We find multiple solutions for semilinear boundary value problems when the corresponding functional exhibits local splitting at zero.

We show how mountain pass methods can be used even when there is no mountain pass or saddle point geometry. The functional can grow on linking subsets. Applications are given.

We prove essential self-adjointness of the operator under weak conditions on the coefficients.

We consider unique continuation theorems for solution of inequalities ¦Δu(x)¦ ⩽ W(x) ¦u(x)¦ with W allowed to be unbounded. We obtain two kinds of results. One allows W ϵ Lploc( R n) with p ⩾ n − 2 for n > 5, p > 1 3 (2n − 1) for n ⩽ 5 . The other requires fW2 to be −Δ-form bounded for all f ϵ C0∞.

For $\lambda \geqq 0$ it is determined precisely which functions $V(x) \geqq 0$ on ${\bf R}^n $ satisfy an inequality of the form \[ (Vu,u) \leqq C_\lambda (V)\left( {\| {\nabla u} \|^2 + \lambda ^2 \| u \|^2 } \right),\quad u \in C_0^\infty \] for some constant $C_\lambda (V)$. The value of the smallest such constant is found. Inequalities of this type are important in the study of the Schrodinger equation. An pplication is given.

A new definition of differentiation for mappings between topological vector spaces is introduced. It does not require the mapping to be defined on a linear manifold, nor does it require it to be continuous. All of the main theorems of differential calculus hold and all other known definitions of differentiation are included. The new definition can be used for singular mappings and those defined on arbitrary sets. Applications are given.

We give sufficient conditions on the potentials to insure that the essential spectrum of such operators coincides with the positive real axis.

It is observed that the perturbation class of an open semigroup in a Banach algebra is a closed two-sided ideal. Certain seminorms on the algebra of bounded operators are introduced; these seminorms induce norms on the quotient algebra modulo the ideal of compact operators. Using these seminorms and an assumption apparently weaker than the metric approximation property it is shown that semiFredholm operators have canonical projections (in the quotient algebra) that are not topological zero divisors. A sufficient condition is found that the converse be true. The special cases of subprojective and superprojective Banach spaces are studied. Some properties of essential spectrum are discussed.