Browsing by Subject "p-Laplacian"

Now showing 1 - 8 of 8

Access status: Restricted ,
Eigenvalue problems for some classes of differential operators
(Data obrony: 2020-12-10) Maciaszek, Michał
Wydział Matematyki Stosowanej
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 ,
On the existence of three solutions for quasilinear elliptic problem
(2012) Goncerz, Paweł
We consider a quasilinear elliptic problem of the type $-\Delta_p u = \lambda (f(u)+\mu g(u))$ in $\Omega$, $u|_{\partial \Omega} =0$, where $\Omega \in \mathbb{R}^N$ is an open and bounded set, $f$, $g$ are continuous real functions on $\mathbb{R}$ and $\lambda , \mu \in \mathbb{R}$. We prove the existence of at least three solutions for this problem using the so called three critical points theorem due to Ricceri.
Access status: Open Access ,
On the solvability of Dirichlet problem for the weighted p-Laplacian
(2012) Szlachtowska, Ewa
The paper investigates the existence and uniqueness of weak solutions for a non-linear boundary value problem involving the weighted $p$-Laplacian. Our approach is based on variational principles and representation properties of the associated spaces.
Access status: Open Access ,
Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions
(Wydawnictwa AGH, 2019) Hai, D. D.; Wang, X.
We prove the existence of positive solutions for the $p$-Laplacian problem $\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}$ where $\phi (s)=|s|^{p-2}s$, $p \gt 1$, $H_{i}:\mathbb{R}\rightarrow\mathbb{R}$ can be nonlinear, $i=1,2$, $f:(0,\infty)\rightarrow \mathbb{R}$ is $p$-superlinear or $p$-sublinear at $\infty$ and is allowed be singular $(\pm\infty)$ at $0$, and $\lambda$ is a positive parameter.
Access status: Open Access ,
Properties of solutions to some weighted p-Laplacian equation
(Wydawnictwa AGH, 2020) Garain, Prashanta
In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by $-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,$ on smooth domain and for varying nonlinearity $f$.
Access status: Open Access ,
Some remarks on the optimization of eigenvalue problems involving the p-Laplacian
(2008) Pielichowski, Wacław
Given a bounded domain $\Omega \subset \mathbb{R}^n$, numbers $p \gt 1$, $\alpha \geq 0$ and $A \in [0,|\Omega |]$, consider the optimization problem: find a subset $D \subset \Omega$, of measure $A$, for which the first eigenvalue of the operator $u\mapsto -\text{div} (|\nabla u|^{p-2}\nabla u)+ \alpha \chi_D |u|^{p-2}u$ with the Dirichlet boundary condition is as small as possible. We show that the optimal configuration $D$ is connected with the corresponding positive eigenfunction u in such a way that there exists a number $t\geq 1$ for which $D=\{u \leq t\}$. We also give a new proof of symmetry of optimal solutions in the case when $\Omega$ is Steiner symmetric and $p=2$.
Access status: Open Access ,
The first eigencurve for a Neumann boundary problem involving p-Laplacian with essentially bounded weights
(Wydawnictwa AGH, 2023) Sanhaji, Ahmed; Dakkak, Ahmed; Moussaoui, Mimoun
This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation $-\Delta_{p} u=\alpha m_{1}|u|^{p-2}u+\beta m_{2}|u|^{p-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^{N}$. We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability.