We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: ( −4pu = λ|u|r−2u+ μ |u| q−2 |x|s u in Ω, u|∂Ω = 0, where λ and μ are two positive parameters and Ω is a smooth bounded domain in Rn containing 0 in its interior. The variational approach requires that 1 < p < n, p ≤ q ≤ p∗(s) ≡ n−s n−pp and p ≤ r ≤ p ∗ ≡...

In this paper, we establish a variant of Ekeland’s variational principle. This result suggest to introduce a generalization of the famous PalaisSmale condition. An example is provided showing how it is used to give the existence of minimizer for functions for which the Palais-Smale condition and the one introduced by Cerami are not satisfied.

Since the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice the solutions are critical points of functionals. If a functional G is semibounded, one can find a Palais-Smale (PS) sequence G(uk) → a, G(uk) → 0. These sequences produce critical points if they have ...

This paper considers bifurcation at the principal eigenvalue of a class of gradient operators which possess the Palais-Smale condition. The existence of the bifurcation branch and the asymptotic nature of the bifurcation is verified by using the compactness in the Palais Smale condition and the order of the nonlinearity in the operator. The main result is applied to estimate the asyptotic behav...

Fixing a complete Riemannian metric g on Rn, we show that a local diffeomorphism f : Rn → Rn is bijective if the height function f · v (standard inner product) satisfies the Palais-Smale condition relative to g for each for each nonzero v ∈ Rn. Our method substantially improves a global inverse function theorem of Hadamard. In the context of polynomial maps, we obtain new criteria for invertibi...

This paper studies the stability properties of the concepts of local compactness introduced by the authors in 1998. We show that all of these concepts are stable for contractive, expansive images and for products. 1. Introduction. Having introduced notions of local compactness, basis local compactness and related measures, and having studied the basic relationship among these concepts in [3], i...

In addition to clinical factors (tumor and node stage) and treatment factors (equivalent radiotherapy dose and chemotherapy regimen), we assessed whether different performances of various tumor volume measurements help predict the pathological complete response (pCR) of locally advanced rectal cancer (LARC) after preoperative concurrent chemoradiotherapy (CCRT). A total of 122 patients with LAR...

We construct local minimum solutions of the semilinear bistable equation by minimizing the corresponding functional near some approximate solutions, assuming that some global minimum solutions are isolated. The key step is to prove with the help of a characterization of Palais-Smale sequences that the functional takes higher values away from the approximate solutions than it does near the appro...