Search results for "Dirichlet problem"
showing 10 items of 79 documents
A singular (p,q)-equation with convection and a locally defined perturbation
2021
Abstract We consider a parametric Dirichlet problem driven by the ( p , q ) -Laplacian and a reaction which is gradient dependent (convection) and the competing effects of two more terms, one a parametric singular term and a locally defined perturbation. We show that for all small values of the parameter the problem has a positive smooth solution.
Critical points in open sublevels and multiple solutions for parameter-depending quasilinear elliptic equations
2014
We investigate the existence of multiple nontrivial solutions of a quasilinear elliptic Dirichlet problem depending on a parameter $\lambda>0$ of the form $$ -\Delta_pu=\lambda f(u)\quad\mbox{in }\ \Omega,\quad u=0\quad\mbox{on }\ \partial\Omega, $$ where $\Omega\subset \mathbb{R}^N$ is a bounded domain, $\Delta_p$, $1 < p < +\infty$, is the $p$-Laplacian, and $f: \mathbb{R}\to \mathbb{R}$ is a continuous function satisfying a subcritical growth condition. More precisely, we establish a variational approach that when combined with differential inequality techniques, allows us to explicitly describe intervals for the parameter $\lambda$ for which the problem under consideration admits nontri…
A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary
2016
We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain $\Omega_{\boldsymbol\varepsilon}$ obtained by making a small hole of size $\varepsilon_1 \varepsilon_2 $ in an open regular subset $\Omega$ of $\mathbb{R}^n$ at distance $\varepsilon_1$ from the boundary $\partial\Omega$. As $\varepsilon_1 \to 0$, the perforation shrinks to a point and, at the same time, approaches the boundary. When $\boldsymbol\varepsilon \to (0,0)$, the size of the hole shrinks at a faster rate than its approach to the boundary. We denote by $u_{\bolds…
Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term
2001
Abstract Our aim in this article is to study the following nonlinear elliptic Dirichlet problem: − div [a(x,u)·∇u]+b(x,u,∇u)=f, in Ω; u=0, on ∂Ω; where Ω is a bounded open subset of RN, with N>2, f∈L m (Ω) . Under wide conditions on functions a and b, we prove that there exists a type of solution for this problem; this is a bounded weak solution for m>N/2, and an unbounded entropy solution for N/2>m⩾2N/(N+2). Moreover, we show when this entropy solution is a weak one and when can be taken as test function in the weak formulation. We also study the summability of the solutions.
Fractional p-Laplacian evolution equations
2016
Abstract In this paper we study the fractional p-Laplacian evolution equation given by u t ( t , x ) = ∫ A 1 | x − y | N + s p | u ( t , y ) − u ( t , x ) | p − 2 ( u ( t , y ) − u ( t , x ) ) d y for x ∈ Ω , t > 0 , 0 s 1 , p ≥ 1 . In a bounded domain Ω we deal with the Dirichlet problem by taking A = R N and u = 0 in R N ∖ Ω , and the Neumann problem by taking A = Ω . We include here the limit case p = 1 that has the extra difficulty of giving a meaning to u ( y ) − u ( x ) | u ( y ) − u ( x ) | when u ( y ) = u ( x ) . We also consider the Cauchy problem in the whole R N by taking A = Ω = R N . We find existence and uniqueness of strong solutions for each of the above mentioned problem…
Quasilinear degenerate parabolic equations in unbounded domains
2004
We prove the existence of bounded solutions of Cauchy-Dirichlet problem associated to a degenerate parabolic equation of second order in divergence form in unbounded domain.
Parabolic Equations Minimizing Linear Growth Functionals: L1-Theory
2004
Let Ω be a bounded set in ℝN with boundary of class C1. We are interested in the problem $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = diva\left( {x,Du} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega \hfill \\ \end{gathered} \right. $$ (1) where ϕ ∈ L1(∂Ω), u0 ∈ L2(Ω) and a(x, ξ) = ∇ξ f(x, ξ, f being a function with linear growth in ‖ξ‖ as ‖ξ‖ → ∞. One of the classical examples is the nonparametric area integrand for which \( f(x,\xi ) = \sqrt {1 + \left\| \xi \right\|^2 } \). Prob…
On Some Properties of the Dirichlet Problem at Resonance
2008
Abstract The boundary value problem at resonance 𝑥″ + 𝑥 = 𝑞 sin 𝑡 + 𝑓(𝑡,𝑥,𝑥′), 𝑥(0) = 0, 𝑥(π) = 0, is considered, where 𝑓 : [0,π] × 𝑹2 → 𝑹 is a bounded Carathéodory function, 𝑞 is a parameter. We state the multiplicity results without assuming that 𝑓 has limits.
A Parametric Dirichlet Problem for Systems of Quasilinear Elliptic Equations With Gradient Dependence
2016
The aim of this article is to study the Dirichlet boundary value problem for systems of equations involving the (pi, qi) -Laplacian operators and parameters μi≥0 (i = 1,2) in the principal part. Another main point is that the nonlinearities in the reaction terms are allowed to depend on both the solution and its gradient. We prove results ensuring existence, uniqueness, and asymptotic behavior with respect to the parameters.
Singular Double Phase Problems with Convection
2020
We consider a nonlinear Dirichlet problem driven by the sum of a $p$ -Laplacian and of a $q$ -Laplacian (double phase equation). In the reaction we have the combined effects of a singular term and of a gradient dependent term (convection) which is locally defined. Using a mixture of variational and topological methods, together with suitable truncation and comparison techniques, we prove the existence of a positive smooth solution.