Browsing by Subject "variational methods"

Now showing 1 - 16 of 16

Access status: Open Access ,
A note on nonlocal discrete problems involving sign-changing Kirchhoff functions
(Wydawnictwa AGH, 2025) Ricceri, Biagio
In this note, we establish a multiplicity theorem for a nonlocal discrete problem of the type \[\begin{cases} -\left(a\sum_{m=1}^{n+1}|x_m-x_{m-1}|^2+b\right)(x_{k+1}-2x_k+x_{k-1})=h_k(x_k), &k=1,\ldots ,n, \\ x_0=x_{n+1}=0 \end{cases}\] assuming $a\gt 0$ and (for the first time) $b\gt 0$.
Access status: Open Access ,
Combined effects for a class of fractional variational inequalities
(Wydawnictwa AGH, 2025) Deng, Shengbing; Luo, Wenshan; Torres Ledesma, César E.
In this paper, we study the existence of a nonnegative weak solution to the following nonlocal variational inequality: $\textcolor{white}\$\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} u (-\Delta)^{{\frac{s}{2}}}(v-u)dx+\int_{\mathbb{R}^N}(1+\lambda M(x))u(v-u)dx \geq \int_{\mathbb{R}^N}f(u)(v-u)dx, \textcolor{white}\$$ for all $v \in\mathbb{K}$, where $s\in (0,1)$ and $M$ is a continuous steep potential well on $\mathbb{R}^N$. Using penalization techniques from del Pino and Felmer, as well as from Bensoussan and Lions, we establish the existence of nonnegative weak solutions. These solutions localize near the potential well $\operatorname{Int}(M^{-1}(0))$.
Access status: Open Access ,
Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball
(Wydawnictwa AGH, 2023) Graef, John R.; Hebboul, Doudja; Moussaoui, Toufik
In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the $p$-Laplacian $-\Big(a+b\int_{\Omega_e}|\nabla u|^p dx\Big)\Delta_p u=\lambda f\left(|x|,u\right),\ x\in \Omega_e,\quad u=0\ \text{on} \ \partial\Omega_e,$ where $\lambda \gt 0$ is a parameter, $\Omega_e = \lbrace x\in\mathbb{R}^N : |x|\gt r_0\rbrace$, $r_{0} \gt 0$, $N \gt p \gt 1$, $\Delta_{p}$ is the $p$-Laplacian operator, and $f\in C(\left[ r_0, +\infty\right)\times\left[0,+\infty\right),\mathbb{R})$ is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of $\lambda$.
Access status: Open Access ,
Existence of three solutions for impulsive nonlinear fractional boundary value problems
(2017) Heidarkhani, Shapour; Ferrara, Massimiliano; Caristi, Giuseppe; Salari, Amjad
In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results.
Access status: Open Access ,
Existence of three solutions for perturbed nonlinear difference equations
(2014) Heidarkhani, Shapour; Moghadam, Mohsen Khaleghi
Using critical point theory, we study the existence of at least three solutions for perturbed nonlinear difference equations with discrete boundary-value condition depending on two positive parameters.
Access status: Open Access ,
Ground states for fractional nonlocal equations with logarithmic nonlinearity
(Wydawnictwa AGH, 2022) Guo, Lifeng; Sun, Yan; Shi, Guannan
In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by $\begin{cases}\mathcal{L}_{K}u(x)+u\log|u|+|u|^{q-2}u=0, & x\in\Omega,\\ u=0, & x\in\mathbb{R}^{n}\setminus\Omega,\end{cases}$ where $2\lt q\lt 2^{*}_s$, $L_{K}$ is a non-local operator, $\Omega$ is an open bounded set of $\mathbb{R}^{n}$ with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.
Access status: Open Access ,
Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type via critical point theory
(2016) Mokhtari, Abdelhak; Moussaoui, Toufik; O'Regan, Donal
In this paper we obtain existence results of $k$ distinct pairs nontrivial solutions for an impulsive boundary value problem of $p(t)$-Kirchhoff type under certain conditions on the parameter $\lambda$.
Access status: Open Access ,
Multiplicity results for perturbed fourth-order Kirchhoff-type problems
(Wydawnictwa AGH, 2017) Tavani, Mohamad Reza Heidari; Afrouzi, Ghasem Alizadeh; Heidarkhani, Shapour
In this paper, we investigate the existence of three generalized solutions for fourth-order Kirchhoff-type problems with a perturbed nonlinear term depending on two real parameters. Our approach is based on variational methods.
Access status: Open Access ,
Normalized solutions for critical Schrödinger equations involving (2,q)-Laplacian
(Wydawnictwa AGH, 2025) Wei, Lulu; Song, Yueqiang
In this paper, we consider the following critical Schrödinger equation involving $(2,q)$-Laplacian: \[\begin{cases} -\Delta u-\Delta_{q} u=\lambda u+\mu |u|^{\gamma-2}u+|u|^{2^*-2}u \quad\text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N} |u|^{2}dx=a^2,\end{cases}\] where $\Delta_q u =\operatorname{div} (|\nabla u|^{q-2}\nabla u)$ is the $q$-Laplacian operator, $\mu, a\gt 0,$ $\lambda\in\mathbb{R}$, $\gamma\in(2,2^*)$, $q\in(\frac{2N}{N+2},2)$ and $N\geq3$. The meaningful and interesting phenomenon is the simultaneous occurrence of $(2,q)$-Laplacian and critical nonlinearity in the above equation. In order to obtain existence of multiple normalized solutions for such equation, we need to make a detailed estimate. More precisely, for the $L^2$-subcritical case, we use the truncation technique, concentration-compactness principle and the genus theory to get the existence of multiple normalized solutions. For the $L^2$-supercritical case, we obtain a couple of normalized solution for the above equation by a fiber map and concentration-compactness principle." /> <meta name="keywords" content="Schrödinger equation, $(2,q)$-Laplacian, variational methods, critical growth, normalized solutions
Access status: Open Access ,
On the dependence on parameters for second order discrete boundary value problems with the p(k)-Laplacian
(2014) Smejda, Joanna; Wieteska, Renata
In this paper we study the existence and the nonexistence of solutions for the boundary value problems of a class of nonlinear second-order discrete equations depending on a parameter. Variational (the mountain pass technique) and non-variational methods are applied.
Access status: Open Access ,
On the Steklov problem involving the p(x)-Laplacian with indefinite weight
(Wydawnictwa AGH, 2017) Ali, Khaled Ben; Ghanmi, Abdeljabbar; Kefi, Khaled
Under suitable assumptions, we study the existence of a weak nontrivial solution for the following Steklov problem involving the $p(x)$-Laplacian $\begin{cases}\Delta_{p(x)}u=a(x)|u|^{p(x)-2}u \quad \text{in }\Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=\lambda V(x)|u|^{q(x)-2}u \quad \text{on }\partial \Omega.\end{cases}$ Our approach is based on min-max method and Ekeland's variational principle.
Access status: Open Access ,
On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative
(Wydawnictwa AGH, 2019) Bełdziński, Michał; Galewski, Marek
In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems.
Access status: Open Access ,
Remarks for one-dimensional fractional equations
(2014) Ferrara, Massimiliano; Molica Bisci, Giovanni
In this paper we study a class of one-dimensional Dirichlet boundary value problems involving the Caputo fractional derivatives. The existence of infinitely many solutions for this equations is obtained by exploiting a recent abstract result. Concrete examples of applications are presented.
Access status: Open Access ,
Some existence results for a nonlocal non-isotropic problem
(Wydawnictwa AGH, 2021) Bentifour, Rachid; Miri, Sofiane El-Hadi
In this paper we deal with the following problem $\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}$ where $\Omega$ is a bounded regular domain in $\mathbb{R}^{N}$. We will assume without loss of generality that $1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}$ and that $f$ and $g$ are non-negative functions belonging to a suitable Lebesgue space $L^{m}(\Omega)$, $1\lt q\lt \overline{p}^{\ast}$, $a \gt 0$, $b \gt 0$ and $0\lt\gamma \lt 1.$
Access status: Open Access ,
Some multiplicity results of homoclinic solutions for second order Hamiltonian systems
(Wydawnictwa AGH, 2020) Barile, Sara; Salvatore, Addolorata
We look for homoclinic solutions $q:\mathbb{R} \rightarrow \mathbb{R}^N$ to the class of second order Hamiltonian systems $-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}$ where $L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}$ and $a,b: \mathbb{R}\rightarrow \mathbb{R}$ are positive bounded functions, $G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}$ are positive homogeneous functions and $f:\mathbb{R}\rightarrow\mathbb{R}^N$. Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if $f\equiv 0$ and the existence of at least three solutions if $f$ is not trivial but small enough.
Access status: Open Access ,
Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications
(Wydawnictwa AGH, 2021) El Amrouss, Abdelrachid; Hammouti, Omar
Let $n\in\mathbb{N}^{*}$, and $N\geq n$ be an integer. We study the spectrum of discrete linear $2n$-th order eigenvalue problems $\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}$ where $\lambda$ is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear $2n$-th order problems by applying the variational methods and critical point theory.