Band 7, Heft 2

We establish the existence of infinitely many solutions for the equation -Δ G u + u = f(ξ,u), ξ ∈ G where Δ G is a sublaplacian on a rational Carnot group G. The function f is assumed to be periodic with respect to a discrete co-compact subgroup of G and satisfy subcritical growth conditions.

In this paper, we prove that if b(x) satisfies some suitable conditions, then -Δu + u = b(x)u p in ℝ N + with the boundary condition u(xʹ, 0) = λg(xʹ) has at least two positive solutions if 0 < λ < λ*, a minimal positive solution if λ = λ* and no positive solution if λ > λ*.

We study the following Dirichlet problem with the p-Laplacian and weights: We show the existence of a positive solution, a negative solution and a sign-changing solution to the problem.

In this paper we prove the existence, in a suitable sense, of a large solution (a solution that blows-up at the boundary) for a class of nonlinear elliptic equations whose model is -Δ p u + u + u|▽u| q = f in Ω, p > 1, p - 1 < q ≤ p and f(x) ∈ L 1 (Ω). The main tool in order to prove it relies on approximating the problem with a more regular one, prove local (i.e. independent from the behavior on the boundary) a priori estimates and local compactness for truncations in W 1,p (Ω). Such scheme of the proof is also applied in order to prove the existence of a solution for equations of the type -Δ p u + u + u|▽u| q = f in ℝ N , where f(x) ∈ L 1 loc (ℝ N ) without any condition at infinity. Moreover, the local summability of the solution depending on the local summability of the datum is studied.

This article is concerned with the existence, uniqueness and numerical approximation of boundary blow up solutions for elliptic PDE’s Δu = f(u), where f satisfies the so-called Keller-Osserman condition. We characterize existence of such solutions for non-monotone f. As an example, we construct an infinite family of boundary blow up solutions for the equation Δu = u 2 (1 + cos u) on a ball. We prove uniqueness (on balls) when f is increasing and convex in a neighborhood of infinity and we discuss and perform some numerical computations to approximate such boundary blow-up solutions.

Given a bounded, open set Ω in ℝ N (N ≥ 3), ψ∈ W 1,p (Ω) (p > N) such that υ̸ + ∈ H 0 1 (Ω) ∩ L ∞ (Ω) and a suitable strictly positive (see (1.4)) function a ∈ L q (Ω) with q > N/2, we prove the existence of positive solution w ∈ H 0 1 (Ω) of some variational inequality with a singular nonlinearity whose typical model is where the set of test functions K 1 consists of all functions υ ∈ H 0 1 (Ω) ∩ L ∞ (Ω) such that υ(x) ≥ ψ(x) a.e. x ∈ Ω and supp (υ - ψ+) ⊂⊂ Ω. Bigger classes of test functions are also studied. We also recover the case in which the variational inequality reduces to an equation.