Search

In this paper, we study the multiplicity results for a second-order Hamiltonian system generated by impulsive effects. Our approach is based on variational methods and critical point theory. [ABSTRACT FROM AUTHOR] more...

This paper deals with the existence and multiplicity of weak solutions for the following Kirchhoff‐type problem M∬ℝ2N1p(x,y)|u(x)−u(y)|p(x,y)|x−y|N+s(x,y)p(x,y)dxdy(−Δ)p(·)s(·)u(x)=f(x,u)inΩ,u(x)=0inℝN\Ω,$$ \left\{\begin{array}{ll}M\left({\iint}_{{\mathrm{\mathbb{R}}}^{2N}}\frac{1}{p\left(x,y\right)}\frac{{\left|u(x)-u(y)\right|}^{p\left(x,y\right)}}{{\left|x-y\right|}^{N+s\left(x,y\right)p\left(x,y\right)}}\kern0.1em dx\kern0.1em dy\right){\left(-\Delta \right)}_{p\left(\cdotp \right)}^{s\left(\cdotp \right)}u(x)=f\left(x,u\right)& \kern0.1em \mathrm{in}\kern0.4em \Omega, \\ {}u(x)=0& \kern0.1em \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}^N\backslash \Omega, \end{array}\right. $$where M$$ M $$ models a Kirchhoff coefficient and (−Δ)p(·)s(·)$$ {\left(-\Delta \right)}_{p\left(\cdotp \right)}^{s\left(\cdotp \right)} $$ is a variable s(·)$$ s\left(\cdotp \right) $$‐order p(·)$$ p\left(\cdotp \right) $$‐fractional Laplace operator, with s(·):ℝ2N→(0,1)$$ s\left(\cdotp \right):{\mathrm{\mathbb{R}}}^{2N}\to \left(0,1\right) $$ and p(·):ℝ2N→(1,∞)$$ p\left(\cdotp \right):{\mathrm{\mathbb{R}}}^{2N}\to \left(1,\infty \right) $$. Here, Ω⊂ℝN$$ \Omega \subset {\mathrm{\mathbb{R}}}^N $$ is a bounded smooth domain with N>s(x,y)p(x,y)$$ N>s\left(x,y\right)p\left(x,y\right) $$ for any (x,y)∈Ω‾×Ω‾$$ \left(x,y\right)\in \overline{\Omega}\times \overline{\Omega} $$. By using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle in two cases when the Carathéodory function f(x,u)$$ f\left(x,u\right) $$ having special structure, we establish conditions ensuring the multiplicity results. [ABSTRACT FROM AUTHOR] more...

Based on the variational methods and critical-point theory, we are concerned with the existence results for a second-order impulsive boundary value problem involving an ordinary differential equation with p(x)-Laplacian operator, and Neumann conditions. [ABSTRACT FROM AUTHOR] more...

In this work, we establish existence results for the following fourth-order Kirchhoff-type elliptic problem with Hardy potential M(∫Ω |∆u| pdx) ∆²ubscript>pu - a |x|p|u|p-2u = λf(x, u), in Ω, u = ∆u = 0, on ∂Ω. Precisely, by using the classical Hardy inequality and critical point theory, we prove the existence of multiple weak solutions for the fourth-order Kirchhoff-type elliptic problem with Hardy potential. [ABSTRACT FROM AUTHOR] more...

We are interested in the existence of multiple solutions for a class of p(x)-curl systems arising in electromagnetism. We work on variable exponent Sobolev spaces and by using critical point theory and the variational method, we investigate the existence of at least one, two, and three solutions to the problem. [ABSTRACT FROM AUTHOR] more...

The existence of at least three weak solutions to a discrete fractional boundary value problem containing a p-Laplacian operator and subject to perturbations is proved using variational methods. Some applications of the main results are presented. The results obtained generalize some recent results on both discrete fractional boundary value problems and p-Laplacian boundary value problems. Examples illustrating the results are given. [ABSTRACT FROM AUTHOR] more...

In this paper we deal with a bifurcation result for the following parametric one-dimensional mean curvature problem:

In this paper, we verify that a general p(x)-Laplacian Neumann problem has at least three weak solutions, which generalizes the corresponding result of the reference [R. A. Mashiyev, Three Solutions to a Neumann Problem for Elliptic Equations with Variable Exponent, Arab. J. Sci. Eng. 36 (2011) 1559-1567]. more...

In this paper, we give some new criteria that guarantee the existence of at least three weak solutions to a p-Hamiltonian boundary value problem generated by impulsive effects. To ensure the existence of these solutions, we use variational methods and critical point theory as our main tools. [ABSTRACT FROM AUTHOR] more...