208 research outputs found
Concentration phenomena for a fractional Schr\"odinger-Kirchhoff type equation
In this paper we deal with the multiplicity and concentration of positive
solutions for the following fractional Schr\"odinger-Kirchhoff type equation
\begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}}
\iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy +
\frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2}
dx\right)[\varepsilon^{2s} (-\Delta)^{s}u+ V(x)u]= f(u) \, \mbox{in}
\mathbb{R}^{3} \end{equation*} where is a small parameter,
, is the fractional Laplacian, is a
Kirchhoff function, is a continuous positive potential and is a
superlinear continuous function with subcritical growth. By using penalization
techniques and Ljusternik-Schnirelmann theory, we investigate the relation
between the number of positive solutions with the topology of the set where the
potential attains its minimum.Comment: Mathematical Methods in the Applied Sciences (2017
A new class of multiple nonlocal problems with two parameters and variable-order fractional -Laplacian
In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff
problem, which involves the -fractional Laplacian equations of variable
order. The problem is stated as follows: \begin{eqnarray*} \left\{
\begin{array}{ll}
M\Big(\sigma_{p(x,y)}(u)\Big)(-\Delta)^{s(\cdot)}_{p(\cdot)}u(x) =\lambda
|u|^{q(x)-2}u\left(\int_\O\frac{1}{q(x)} |u|^{q(x)}dx
\right)^{k_1}+\beta|u|^{r(x)-2}u\left(\int_\O\frac{1}{r(x)} |u|^{r(x)}dx
\right)^{k_2}
\quad \mbox{in }\Omega,
\\ u=0 \quad \mbox{on }\partial\Omega, \end{array} \right. \end{eqnarray*}
where the nonlocal term is defined as Here, represents a bounded smooth
domain with at least . The function is given by , where , , and . The parameters ,
, and are real parameters, while the variables ,
, , and are continuous and can change with respect to
. To tackle this problem, we employ some new methods and variational
approaches along with two specific methods, namely the Fountain theorem and the
symmetric Mountain Pass theorem. By utilizing these techniques, we establish
the existence and multiplicity of solutions for this problem separately in two
distinct cases: when and when . To the best of our knowledge, these
results are the first contributions to research on the variable-order
-fractional Laplacian operator.Comment: 21 page
Perturbed nonlocal fourth order equations of Kirchhoff type with Navier boundary conditions
Abstract We investigate the existence of multiple solutions for perturbed nonlocal fourth-order equations of Kirchhoff type under Navier boundary conditions. We give some new criteria for guaranteeing that the perturbed fourth-order equations of Kirchhoff type have at least three weak solutions by using a variational method and some critical point theorems due to Ricceri. We extend and improve some recent results. Finally, by presenting two examples, we ensure the applicability of our results
Nonzero positive solutions of fractional Laplacian systems with functional terms
We study the existence of non-zero positive solutions of a class of systems
of differential equations driven by fractional powers of the Laplacian. Our
approach is based on the notion of fixed point index, and allows us to deal
with non-local functional weights and functional boundary conditions. We
present two examples to shed light on the type of functionals and growth
conditions that can be considered with our approach
Existence and asymptotic analysis of positive solutions for a singular fractional differential equation with nonlocal boundary conditions
In this paper, we focus on the existence and asymptotic analysis of positive solutions for a class of singular fractional differential equations subject to nonlocal boundary conditions. By constructing suitable upper and lower solutions and employing Schauder’s fixed point theorem, the conditions for the existence of positive solutions are established and the asymptotic analysis for the obtained solution is carried out. In our work, the nonlinear function involved in the equation not only contains fractional derivatives of unknown functions but also has a stronger singularity at some points of the time and space variables
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