Existence of bounded solutions for nonlinear elliptic equations in unbounded domains (original) (raw)

Existence of bounded solutions for nonlinear elliptic uni-lateral problems

Annali Di Matematica Pura Ed Applicata - ANN MAT PUR APPL, 1988

In this paper we study the existence of bounded weak solutionsfor some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the plaplacian operator, and the lower order terms, which depend on the solution u and its gradient ∇u, have a power growth of order p − 1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.

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.

A priori estimates for some elliptic equations involving the p-Laplacian

Nonlinear Analysis: Real World Applications, 2018

We consider the Dirichlet problem for positive solutions of the equation −∆ p (u) = f (u) in a convex, bounded, smooth domain Ω ⊂ R N , with f locally Lipschitz continuous. We provide sufficient conditions guarantying L ∞ a priori bounds for positive solutions of some elliptic equations involving the p-Laplacian and extend the class of known nonlinearities for which the solutions are L ∞ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains.

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.

Nonlinear elliptic equations having a gradient term with natural growth

Journal de Mathématiques Pures et Appliquées, 2006

In this paper, we study a class of nonlinear elliptic Dirichlet problems whose simplest model example is: − p u = g(u)|∇u| p + f, in Ω, u = 0, on ∂Ω. (1) Here Ω is a bounded open set in R N (N 2), p denotes the so-called p-Laplace operator (p > 1) and g is a continuous real function. Given f ∈ L m (Ω) (m > 1), we study under which growth conditions on g problem (1) admits a solution. If m N/p, we prove that there exists a solution under assumption (3) (see below), and that it is bounded when m > N/p; while if 1 < m < N/p and g satisfies the condition (4) below, we prove the existence of an unbounded generalized solution. Note that no smallness condition is asked on f. Our methods rely on a priori estimates and compactness arguments and are applied to a large class of equations involving operators of Leray-Lions type. We also make several examples and remarks which give evidence of the optimality of our results.

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.