Related papers: A Generalized Palais-Smale Condition in the Fr\'ec…

We introduce a class of functionals on the space of rapidly decreasing sequences $s$, called $\mathcal{F}_s$-functionals, defined as decomposable sums of quadratic and convex terms with quadratic growth. We prove that such functionals…

We give sufficient conditions for a $ C^1_c $-local diffeomorphism between Fr\'{e}chet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fr\'{e}chet spaces. As a consequence, we…

In this paper we study a quasilinear elliptic problem whose functional satisfies a weak version of the well known Palais-Smale condition. An existence result is proved under general assumptions on the nonlinearities.

Without compactness assumptions, we prove some abstract results which show that a $C^{1}$ functional $I:X\rightarrow \mathbb{R}$ on a Banach space $X$ admits bounded Palais-Smale sequences provided that it exhibits some geometric structure…

We show that the Hamiltonian action satisfies the Palais-Smale condition over a "mixed regularity" space of loops in cotangent bundles, namely the space of loops with regularity $H^s$, $s\in (\frac 12, 1)$, in the base and $H^{1-s}$ in the…

Let f be local diffeomorphism between real Banach spaces. We prove that if the locally Lipschitz functional F(x)=1/2|f(x)-y|^2 satisfies the Chang Palais-Smale condition for all y in the target space of f, then f is a norm-coercive global…

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients…

Approximate necessary optimality conditions in terms of Fr\'echet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's…

We use the framework of a type of abstract convexity ($\Phi_{lsc}$-convexity) to investigate properties of lower semicontinuous quadratically minorized functions in Hilbert spaces. A new result, which states that, for every local…

In this paper we fully characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem. We also establish sufficient criteria with respect to the…

We consider a class of scalar field equations with anisotropic nonlocal nonlinearities. We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to this variable exponent setting, and use it to prove that the…

We establish lower semicontinuity results for perimeter functionals with measure data on $\mathbb{R}^n$ and deduce the existence of minimizers to these functionals with Dirichlet boundary conditions, obstacles, or volume-constraints. In…

We study dynamic minimization problems of the calculus of variations with generalized Lagrangian functionals that depend on a general linear operator $K$ and defined on bounded-time intervals. Under assumptions of regularity, convexity and…

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $\left(\varphi_1, \varphi_2\right)-$convex function $g, $ with arbitrarily small norm, such that $f + g…

We prove minimax theorems for lower semicontinuous functions defined on a Hilbert space. The main tool is the theory of $\Phi$-convex functions and sufficient and necessary conditions for the minimax equality to hold for $\Phi$-convex…

This paper is concerned with index pairs in the sense of Conley index theory for flows relative to pseudo-gradient vector fields for $C^1$-functions satisfying Palais-Smale condition. We prove a deformation theorem for such index pairs to…

We perform a systematic variational method for functionals depending on eigenvalues of Riemannian manifolds. It is based on a new concept of Palais Smale sequences that can be constructed thanks to a generalization of classical min-max…

We prove an implicit function theorem for Keller C^k_c-maps from arbitrary real or complex topological vector spaces to Frechet spaces, imposing only a certain metric estimate on the partial differentials. As a tool, we show the…

We prove that Palais-Smale sequences for Liouville type functionals on closed surfaces are precompact whenever they satisfy a bound on their Morse index. As a byproduct, we obtain a new proof of existence of solutions for Liouville type…

In this paper, we point out a very flexible scheme within which a strict minimax inequality occurs. We then show the fruitfulness of this approach presenting a series of various consequences. Here is one of them: Let $Y$ be a…