Browsing by Subject "variational method"

Now showing 1 - 5 of 5

Access status: Open Access ,
A note on a fourth order discrete boundary value problem
(2012) Galewski, Marek; Smejda, Joanna
Using variational methods we investigate the existence of solutions and their dependence on parameters for certain fourth order difference equations.
Access status: Open Access ,
Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator
(2014) Tsouli, Najib; Darhouche, Omar
In this paper we study the following nonlinear boundary-value problem $-\Delta_{p(x)} u=\lambda f(x,u) \quad \text{ in } \Omega,$ $|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}+\beta(x)|u|^{p(x)-2}u=\mu g(x,u) \quad \text{ on } \partial\Omega,$ where $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $\frac{\partial u}{\partial\nu}$ is the outer unit normal derivative on $\partial\Omega$, $\lambda, \mu$ are two real numbers such that $\lambda^{2}+\mu^{2}\neq0$, $p$ is a continuous function on $\overline{\Omega}$ with $\inf_{x\in \overline{\Omega}} p(x)\gt 1$, $\beta\in L^{\infty}(\partial\Omega)$ with $\beta^{-}:=\inf_{x\in \partial\Omega}\beta(x)\gt 0$ and $f : \Omega\times\mathbb{R}\rightarrow \mathbb{R}$, $g : \partial\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ are continuous functions. Under appropriate assumptions on $f$ and $g$, we obtain the existence and multiplicity of solutions using the variational method. The positive solution of the problem is also considered.
Access status: Open Access ,
Existence of three solutions for impulsive multi-point boundary value problems
(2017) Bohner, Martin; Heidarkhani, Shapour; Salari, Amjad; Caristi, Giuseppe
This paper is devoted to the study of the existence of at least three classical solutions for a second-order multi-point boundary value problem with impulsive effects. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. Also by presenting an example, we ensure the applicability of our results.
Access status: Open Access ,
Properties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential
(Wydawnictwa AGH, 2024) Torres Ledesma, César E.; Gutierrez, Hernán C.; Rodríguez, Jesús A.; Bonilla, Manuel M.
In this article we consider the following fractional semilinear elliptic equation $(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,$ where $s\in (0,1)$, $N\gt 2s$, $\sigma\in (0,\frac{2s}{N-2s})$ and $\omega\in (0, \lambda_1)$. By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result
Access status: Open Access ,
Sobolev norm estimates of solutions for the sublinear Emden-Fowler equation
(2013) Kajikiya, Ryuji
We study the sublinear Emden-Fowler equation in small domains. As the domain becomes smaller, so does any solution. We investigate the convergence rate of the Sobolev norm of solutions as the volume of the domain converges to zero. The result is obtained by estimating the first eigenvalue of the Laplacian with the help of the variational method.