Browsing by Subject "weak solutions"

Now showing 1 - 6 of 6

Access status: Open Access ,
A Neumann boundary value problem for a class of gradient systems
(2014) Pan, Wen-Wu; Li, Lin
In this paper we study a class of two-point boundary value systems. Using very recent critical points theorems, we establish the existence of one non-trivial solution and in?nitely many solutions of this problem, respectively.
Access status: Open Access ,
Existence and regularity of solutions for hyperbolic functional differential problems
(2014) Kamont, Zdzisław
A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper.
Access status: Open Access ,
Exponential decay of solutions to a class of fourth-order nonlinear hyperbolic equations modeling the oscillations of suspension bridges
(Wydawnictwa AGH, 2022) Liu, Yang; Yang, Chao
This paper is concerned with a class of fourth-order nonlinear hyperbolic equations subject to free boundary conditions that can be used to describe the nonlinear dynamics of suspension bridges.
Access status: Open Access ,
Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators
(2016) Kong, Lingju
We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a $p(x)$-biharmonic operator $\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $p\in C(\overline{\Omega})$, $\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$ is the $p(x)$-biharmonic operator, and $\lambda\gt 0$ is a parameter. We establish sufficient conditions under which there exists a positive number $\lambda^{*}$ such that the above problem has at least two nontrivial weak solutions for each $\lambda\gt\lambda^{*}$. Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces $L^{p(x)}(\Omega)$ and $W^{k,p(x)}(\Omega)$.
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 ,
Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions
(Wydawnictwa AGH, 2023) Godoy, Tomas
Let $\Omega$ be a $C^2$ bounded domain in $\mathbb{R}^{n}$ such that $\partial\Omega=\Gamma_{1}\cup\Gamma_{2}$, where $\Gamma_1$ and $\Gamma_2$ are disjoint closed subsets of $\partial \Omega$, and consider the problem $-\Delta u=g(\cdot,u)$ in $\Omega$, $u=\tau$ on $\Gamma_1$, $\frac{\partial u}{\partial\nu}=\eta$ on $\Gamma_2$, where $0\leq\tau\in W^{\frac{1}{2},2}(\Gamma_{1})$, $\eta\in(H_{0,\Gamma_{1}}^{1}(\Omega))^{\prime}$, and $g:\Omega \times(0,\infty)\rightarrow\mathbb{R}$ is a nonnegative Carathéodory function. Under suitable assumptions on $g$ and $\eta$ we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow $g$ to be singular at $s=0$ and also at $x \in S$ for some suitable subsets $S\subset\overline{\Omega}$. The Dirichlet problem $-\Delta u=g(\cdot,u)$ in $\Omega$, $u=\sigma$ on $\partial \Omega$ is also studied in the case when $0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)$.