Lower and upper solutions for elliptic problems in nonsmooth domains (original) (raw)

Existence and Boundedness of Solutions for Elliptic Equations in General Domains

Advances in Science, Technology and Engineering Systems Journal, 2017

This article is devoted to study the existence of solutions for the strongly nonlinear p(x)-elliptic problem: −∆ p(x) (u) + α 0 |u| p(x)−2 u = d(x) |∇u| p(x) |u| p(x) + 1 + f − div g(x) in Ω, u ∈ W 1,p(x) 0 (Ω), where Ω is an open set of R N , possibly of infinite measure, also we will give some regularity results for these solutions.

On Elliptic Problems in Domains with Unbounded Boundary

Proceedings of the Edinburgh Mathematical Society, 2006

The paper deals with problems of the type −Deltau+a(x)u=∣u∣p−2u-\Delta u+a(x)u=|u|^{p-2}u−Deltau+a(x)u=∣u∣p−2u, ugt0u\gt0ugt0, with zero Dirichlet boundary condition on unbounded domains in mathbbRN\mathbb{R}^NmathbbRN, Ngeq2N\geq2Ngeq2, with a(x)geqcgt0a(x)\geq c\gt0a(x)geqcgt0, pgt2p\gt2pgt2 and plt2N/(N−2)p\lt2N/(N-2)plt2N/(N−2) if Ngeq3N\geq3Ngeq3. The lack of compactness in the problem, related to the unboundedness of the domain, is analysed. Moreover, if the potential a(x)a(x)a(x) has kkk suitable ‘bumps’ and the domain has hhh suitable ‘holes’, it is proved that the problem has at least 2(h+k)2(h+k)2(h+k) positive solutions ($h$ or kkk can be zero). The multiplicity results are obtained under a geometric assumption on varOmega\varOmegavarOmega at infinity which ensures the validity of a local Palais–Smale condition.

Nonlinear boundary conditions for elliptic equations

Electronic Journal of Differential …, 2005

This work is devoted to the study of the elliptic equation ∆u = f (x, u) in a bounded domain Ω ⊂ R n with a nonlinear boundary condition. We obtain various existence results applying coincidence degree theory and the method of upper and lower solutions.

Existence of Strictly Positive Solutions for Sublinear Elliptic Problems in Bounded Domains

Advanced Nonlinear Studies, 2014

Let Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k1ξp ≤ f (ξ) ≤ k2ξp for all ξ ≥ 0 and some k1, k2 > 0 and p ∈ (0, 1). We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form −Δu = m(x) f (u) in Ω, u = 0 on ∂Ω.

A note on the Maximum Principle and the Iteration Method for elliptic equations

2021

We use an iteration procedure propped up by a a classical form of the maximum principle to show the existence of solutions to a nonlinear Poisson equation with Dirichlet boundary conditions. These methods can be applied to the case of special unbounded domains, and can be adapted to show the existence of nontrivial solutions to systems, which we show via some examples.

Bounded Solutions for Nonlinear Elliptic Equations in Unbounded Domains

Journal of Applied Analysis, 2000

In this paper, we prove L ∞-regularity for solutions of some nonlinear elliptic equations with degenerate coercivity whose prototype is      −div(1 (1+|u|) θ(p−1) |∇u| p−2 ∇u) = f in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in IR N , N ≥ 2, 1 < p < N , θ is a real such that 0 ≤ θ ≤ 1 and f ∈ L N p log α L with some α > 0.