We establish some existence results for a class of critical $N$-Laplacian problems in a bounded domain in $\mathbb {R}^{N}$. In the absence of a suitable direct sum decomposition of the underlying Sobolev space to which the classical linking theorem can be applied, we use an abstract linking theorem based on the $\mathbb {Z}_2$-cohomological index to obtain a non-trivial critical point.

This article aims to test whether the Theta System of Reinhart (1996, 2000, 2001, 2002) can account for the puzzles associated with psychological verbs in Polish. The first puzzle, called argument linking, relates to the mapping of the Experiencer onto a subject or an object position. The second puzzle, referred to as case linking, concerns the fact that Experiencers may be marked for different cases in the same sentence position. The analysis of Object Experiencer (OE)/Subject Experiencer (SE) alternations in Polish carried out in this article demonstrates that the predictions of the Theta System about Experiencer argument linking are borne out by the Polish data. SE alternants of eventive OE verbs in Polish show unergative properties, which directly follows from the mechanisms of the Theta System. However, the Theta System faces problems when confronted with dyadic OE verbs with dative Experiencers. The model predicts that dative Experiencers are merged internally, as a part of an unaccusative structure. This prediction is untenable for Polish, because dative Experiencers of dyadic predicates show some characteristics of external arguments, and hence must merge externally. Consequently, the conclusion drawn is that the Theta System can provide solutions to some, but not all, of the argument and case-linking puzzles associated with Polish Experiencers.

The set N of all null geodesics of a globally hyperbolic $(d + 1)$-dimensional spacetime $(M, g)$ is naturally a smooth $(2d - 1)$-dimensional contact manifold. The sky of an event x in M is the subset X of N consisting of all null geodesics through x, and is an embedded Legendrian submanifold of N diffeomorphic to $S^{(d - 1)}$. It was conjectured by Low that for $d = 2$ two events x and y are causally related if and only if X and Y are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for $d = 3$ smooth linking should be replaced with Legendrian linking.

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V(x)u=K(x)|u|^{2^\ast-2}u+g(x, u), u\in W^{1,2}({\bf R}^N)$, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V; K>0. If $0<g(x, u)u\leq c|u|^{2^\ast}$ for an appropriate constant c, we show that this equation has a nontrivial solution.

In this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.