Browsing by Subject "Dirichlet problem"

Now showing 1 - 5 of 5

Access status: Open Access ,
Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus
(2015) Dridi, Safa; Khamessi, Bilel
In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: $-\Delta u=q(x)u^{\sigma }\;\text{in}\;\Omega,\quad u_{|\partial\Omega}=0.$. Here $\Omega$ is an annulus in $\mathbb{R}^{n}$, $n\geq 3$, $\sigma \lt 1$ and $q$ is a positive function in $\mathcal{C}_{loc}^{\gamma }(\Omega )$, $0\lt\gamma \lt 1$, satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory
Access status: Open Access ,
Existence of solutions of the Dirichlet problem for an infinite system of nonlinear differential-functional equations of elliptic type
(2005) Zabawa, Tomasz
The Dirichlet problem for an inﬁnite weakly coupled system of semilinear differential-functional equations of elliptic type is considered. It is shown the existence of solutions to this problem. The result is based on Chaplygin’s method of lower and upper functions.
Access status: Open Access ,
Positive solutions with specific asymptotic behavior for a polyharmonic problem on Rn
(2015) Dhifli, Abdelwaheb
This paper is concerned with positive solutions of the semilinear polyharmonic equation $(-\Delta)^{m} u = a(x){u}^{\alpha}$ on $\mathbb{R}^{n}$, where $m$ and $n$ are positive integers with $n\gt 2m$, $\alpha\in (-1,1)$. The coefficient a is assumed to satisfy $a(x)\approx{(1+|x|)}^{-\lambda}L(1+|x|)\quad \text{for}\quad x\in \mathbb{R}^{n},$ where $\lambda\in [2m,\infty)$ and $L\in C^{1}([1,\infty))$ is positive with $\frac{tL'(t)}{L(t)}\longrightarrow 0$ as $t\longrightarrow \infty$; if $\lambda=2m$, one also assumes that $\int_{1}^{\infty}t^{-1}L(t)dt\lt \infty$. We prove the existence of a positive solution $u$ such that $u(x)\approx{(1+|x|)}^{-\widetilde{\lambda}}\widetilde{L}(1+|x|) \quad\text{for}\quad x\in \mathbb{R}^{n},$ with $\widetilde{\lambda}:=\min(n-2m,\frac{\lambda-2m}{1-\alpha})$ and a function $\widetilde{L}$, given explicitly in terms of $L$ and satisfying the same condition at infinity. (Given positive functions $f$ and $g$ on $\mathbb{R}^{n}$, $f\approx g$ means that $c^{-1}g\leq f\leq cg$ for some constant $c\gt 1$.)
Access status: Open Access ,
Positivity preserving results for a biharmonic equation under Dirichlet boundary conditions
(2014) Omrane, Hanen Ben; Khenissy, Saïma
We prove a dichotomy result giving the positivity preserving property for a biharmonic equation with Dirichlet boundary conditions arising in MEMS models. We adapt some ideas in [H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997), 589–626].
Access status: Open Access ,
Pseudo-differential equations and conical potentials: 2-dimensional case
(Wydawnictwa AGH, 2019) Vasilev, Vladimir Borisovič
We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given.