1,604 research outputs found
Sequences of weak solutions for fractional equations
This work is devoted to study the existence of infinitely many weak solutions
to nonlocal equations involving a general integrodifferential operator of
fractional type. These equations have a variational structure and we find a
sequence of nontrivial weak solutions for them exploiting the
-symmetric version of the Mountain Pass Theorem. To make the
nonlinear methods work, some careful analysis of the fractional spaces involved
is necessary. As a particular case, we derive an existence theorem for the
fractional Laplacian, finding nontrivial solutions of the equation
\left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & {\mbox{in}} \Omega\\ u=0 &
{\mbox{in}} \erre^n\setminus \Omega. \end{array} \right. As far as we know,
all these results are new and represent a fractional version of classical
theorems obtained working with Laplacian equations
Yamabe-type equations on Carnot groups
This article is concerned with a class of elliptic equations on Carnot groups
depending of one real positive parameter and involving a critical nonlinearity.
As a special case of our results we prove the existence of at least one
nontrivial solution for a subelliptic critical equation defined on a smooth and
bounded domain of the {Heisenberg group} . Our approach is based on pure variational methods and locally
sequentially weakly lower semicontinuous arguments
Existence Results for a critical fractional equation
We are concerned with existence results for a critical problem of
Brezis-Nirenberg Type involving an integro-differential operator. Our study
includes the fractional Laplacian. Our approach still applies when adding small
singular terms. It hinges on appropriate choices of parameters in the
mountain-pass theore
Existence results for nonlinear elliptic problems on fractal domains
Some existence results for a parametric Dirichlet problem defined on the
Sierpi\'nski fractal are proved. More precisely, a critical point result for
differentiable functionals is exploited in order to prove the existence of a
well determined open interval of positive eigenvalues for which the problem
admits at least one non-trivial weak solution
Nonlinear equations involving the square root of the Laplacian
In this paper we discuss the existence and non-existence of weak solutions to
parametric fractional equations involving the square root of the Laplacian
in a smooth bounded domain ()
and with zero Dirichlet boundary conditions. Namely, our simple model is the
following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda
f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega.
\end{array}\right. \end{equation*} The existence of at least two non-trivial
-bounded weak solutions is established for large value of the
parameter requiring that the nonlinear term is continuous,
superlinear at zero and sublinear at infinity. Our approach is based on
variational arguments and a suitable variant of the Caffarelli-Silvestre
extension method
Multiple solutions of nonlinear equations involving the square root of the Laplacian
In this paper we examine the existence of multiple solutions of parametric
fractional equations involving the square root of the Laplacian in a
smooth bounded domain () and with
Dirichlet zero-boundary conditions, i.e. \begin{equation*} \left\{
\begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on
} \partial\Omega. \end{array}\right. \end{equation*} The existence of at least
three -bounded weak solutions is established for certain values of
the parameter requiring that the nonlinear term is continuous and
with a suitable growth. Our approach is based on variational arguments and a
variant of Caffarelli-Silvestre's extension method
Multiplicity results for elliptic Kirchhoff-type problems
Abstract
The aim of this paper is to establish the existence of multiple solutions for a perturbed Kirchhoff-type problem depending on two real parameters. More precisely, we show that an appropriate oscillating behaviour of the nonlinear part, even under small perturbations, ensures the existence of at least three nontrivial weak solutions. Our approach combines variational methods with properties of nonlocal fractional operators
Clinical relevance and treatment of nonautoimmune anemia in chronic lymphocytic leukemia
Anemia has an unfavorable impact on quality of life in chronic lymphocytic leukemia (CLL), increases the likelihood of receiving blood transfusions, and eventually has a negative impact on overall survival. Although discrepancies in perception of health-related quality of life between doctors and patients lead to the undertreatment of anemia, CLL patients undergoing chemotherapy who have a hemoglobin level <10 g/dL should be considered for treatment with erythropoiesis-stimulating agents. For hemoglobin values of 10–12 g/dL, the role of performance status and comorbidities should not be underestimated. In this setting, the evaluation of physical fitness using the Cumulative Illness Rating Scale should help physicians to identify those patients with hemoglobin levels of 10–12 g/dL who are suitable for therapy with erythropoiesis-stimulating agents. Finally, the increasing use of aggressive approaches to therapy should encourage physicians towards appropriate management of chemotherapy-induced anemia in CLL patients
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