Selected new aspects of the calculus of variations in the large

Abstract

Abstract. We discuss some of the recent developments in variational methods while emphasizing new applications to nonlinear problems. We touch on several issues: (i) the formulation of variational set-ups which provide more information on the location of critical points and therefore on the qualitative properties of the solutions of corresponding Euler-Lagrange equations; (ii) the relationships between the energy of variationally generated solutions, their Morse indices, and the Hausdorff measure of their nodal sets; (iii) the gluing of several topological obstructions; (iv) the preservation of critical levels after deformation of functionals; (v) and the various ways to recover compactness in certain borderline variational problems.

Bifurcation theorems for nonlinear problems with lack of compactness

“…We refer moreover to the recent papers [2] and [4] for a complete survey of the topic. The authors of [4] deal with the problem…”

Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential

“…A. Ferrero and F. Gazzola [8] also obtained some results for problem (1.1). For other relevant papers see [1,3,6,7,10,12], and the references therein.…”

Nonautonomous second order Hamiltonian systems

“…For the homogeneous case, Ben-Naoum, Troestler and Willem [Ben-Naoum et al 1994] proved the existence of a nonconstant solution. For the case T = 2π , Theorem 1.7, with substantially stronger hypotheses, was proved by Nirenberg; see [Ekeland and Ghoussoub 2002]. Among other things, they assumed…”