10,474 research outputs found
Multiple solutions for the laplace operator with critical growth
The aim of this paper is to extend previous results regarding the
multiplicity of solutions for quasilinear elliptic problems with critical
growth to the variable exponent case.
We prove, in the spirit of \cite{DPFBS}, the existence of at least three
nontrivial solutions to the following quasilinear elliptic equation
in a smooth bounded domain
of with homogeneous Dirichlet boundary conditions on
. We assume that , where
is the critical Sobolev exponent for variable exponents
and is the
laplacian. The proof is based on variational arguments and the extension
of concentration compactness method for variable exponent spaces
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
Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities
We study the stability and exact multiplicity of periodic solutions of the
Duffing equation with cubic nonlinearities. We obtain sharp bounds for h such
that the equation has exactly three ordered T-periodic solutions. Moreover,
when h is within these bounds, one of the three solutions is negative, while
the other two are positive. The middle solution is asymptotically stable, and
the remaining two are unstable.Comment: Keywords: Duffing equation; Periodic solution; Stabilit
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