844 research outputs found
Hamiltonian elliptic systems: a guide to variational frameworks
Consider a Hamiltonian system of type where is a power-type nonlinearity, for instance , having subcritical growth, and is a bounded domain
of , . The aim of this paper is to give an overview of
the several variational frameworks that can be used to treat such a system.
Within each approach, we address existence of solutions, and in particular of
ground state solutions. Some of the available frameworks are more adequate to
derive certain qualitative properties; we illustrate this in the second half of
this survey, where we also review some of the most recent literature dealing
mainly with symmetry, concentration, and multiplicity results. This paper
contains some original results as well as new proofs and approaches to known
facts.Comment: 78 pages, 7 figures. This corresponds to the second version of this
paper. With respect to the original version, this one contains additional
references, and some misprints were correcte
Mountain pass solutions for the fractional Berestycki-Lions problem
We investigate the existence of least energy solutions and infinitely many
solutions for the following nonlinear fractional equation (-\Delta)^{s} u =
g(u) \mbox{ in } \mathbb{R}^{N}, where , ,
is the fractional Laplacian and is an
odd function satisfying Berestycki-Lions type
assumptions. The proof is based on the symmetric mountain pass approach
developed by Hirata, Ikoma and Tanaka in \cite{HIT}. Moreover, by combining the
mountain pass approach and an approximation argument, we also prove the
existence of a positive radially symmetric solution for the above problem when
satisfies suitable growth conditions which make our problem fall in the so
called "zero mass" case
Double-phase problems with reaction of arbitrary growth
We consider a parametric nonlinear nonhomogeneous elliptic equation, driven
by the sum of two differential operators having different structure. The
associated energy functional has unbalanced growth and we do not impose any
global growth conditions to the reaction term, whose behavior is prescribed
only near the origin. Using truncation and comparison techniques and Morse
theory, we show that the problem has multiple solutions in the case of high
perturbations. We also show that if a symmetry condition is imposed to the
reaction term, then we can generate a sequence of distinct nodal solutions with
smaller and smaller energies
Lectures on the free period Lagrangian action functional
In this expository article we study the question of the existence of periodic
orbits of prescribed energy for classical Hamiltonian systems on compact
configuration spaces. We use a variational approach, by studying how the
behavior of the free period Lagrangian action functional changes when the
energy crosses certain values, known as the Ma\~n\'e critical values.Comment: This expository article is the outcome of two series of lectures that
the author gave at two summer schools at the Korea Institute for Advanced
Study of Seul in 2010 and at the Universit\'e de Neuch\^atel in 201
Robin problems with a general potential and a superlinear reaction
We consider semilinear Robin problems driven by the negative Laplacian plus
an indefinite potential and with a superlinear reaction term which need not
satisfy the Ambrosetti-Rabinowitz condition. We prove existence and
multiplicity theorems (producing also an infinity of smooth solutions) using
variational tools, truncation and perturbation techniques and Morse theory
(critical groups)
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