26 research outputs found
Existence and concentration of solutions for a class of biharmonic equations
Some superlinear fourth order elliptic equations are considered. Ground
states are proved to exist and to concentrate at a point in the limit. The
proof relies on variational methods, where the existence and concentration of
nontrivial solutions are related to a suitable truncated equation.Comment: 18 page
Singular -biharmonic problems involving the Hardy-Sobolev exponent
This paper is concerned with existence results for the singular
-biharmonic problem involving the Hardy potential and the critical
Hardy-Sobolev exponent. More precisely, by using variational methods combined
with the Mountain pass theorem and the Ekeland variational principle, we
establish the existence and multiplicity of solutions. To illustrate the
usefulness of our results, an illustrative example is also presented
Caffarelli-Kohn-Nirenberg inequality for biharmonic equations with inhomogeneous term and Rellich potential
In this article, multiplicity of nontrivial solutions for an inhomogeneous singular biharmonic equation with Rellich potential are studied. Firstly, a negative energy solution of the studied equations is achieved via the Ekeland’s variational principle and Caffarelli–Kohn–Nirenberg inequality. Then by applying Mountain pass theorem lack of Palais–Smale conditions, the second solution with positive energy is also obtained
Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian
We consider a partial differential equation that arises in the coarse-grained
description of epitaxial growth processes. This is a parabolic equation whose
evolution is governed by the competition between the determinant of the Hessian
matrix of the solution and the biharmonic operator. This model might present a
gradient flow structure depending on the boundary conditions. We first extend
previous results on the existence of stationary solutions to this model for
Dirichlet boundary conditions. For the evolution problem we prove local
existence of solutions for arbitrary data and global existence of solutions for
small data. By exploiting the boundary conditions and the variational structure
of the equation, according to the size of the data we prove finite time blow-up
of the solution and/or convergence to a stationary solution for global
solutions
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