This note deals with the approximation, by a P1 finite element method with numerical integration, of solution curves of a semilinear problem. Because of both mixed boundary conditions and geometrical properties of the domain, some of the solutions do not belong to H2. So, classical results for convergence lead to poor estimates. We show how to improve such estimates with the use of weighted Sobolev spaces together with a mesh “a priori adapted” to the singularity. For the H1 or L2-norms, we...

We give the precise behaviour of some solutions of a nonlinear elliptic B.V.P. in a bounded domain when a parameter approaches an eigenvalue of the principal part. If the nonlinearity has some regularity and the domain is for example convex, we also prove a nonlinear version of Courant’s Nodal theorem.

We give the precise behaviour of some solutions of a nonlinear elliptic B.V.P. in a bounded domain when a parameter approaches an eigenvalue of the principal part. If the nonlinearity has some regularity and the domain is for example convex, we also prove a nonlinear version of Courant's Nodal theorem.

In this paper we consider the $\varphi$-Laplacian problem with Dirichlet boundary condition, $$-\mathrm{div}\left(\varphi \left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right){=\lambda g(·)\varphi \left(u\right)\text{in}\Omega ,\lambda \in ℝ\text{and}u|}_{\partial \Omega }=0.$$ The term $\varphi$ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^{\infty }\left(\Omega \right)$ and $\Omega \subseteq {ℝ}^N$ is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both a minimizer...

In the recent years, many results have been established on positive solutions for boundary value problems of the form $-div\left(\right|\nabla u\left(x\right){|}^{p-2}\nabla u\left(x\right))=\lambda f\left(u\left(x\right)\right)$ in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).