Sergio Solimini

Some remarks on the number of solutions of some nonlinear elliptic problems

Abstract This paper studies the multiplicity of solutions of the problem: depending on the parameter t for certain terms ϕ. The main hypothesis on f is that, setting f ± = lim s → ± ∞ f ( x , s ) s , in the interval ]f−, f+[ there are eigenvalues of −Δ with the Dirichlet homogeneous boundary conditions on ∂Ω. Studying the « bifurcation from infinity » of the solutions of the problem, multiplicity or sharp multiplicity results are obtained in the case that such eigenvalues are the first two or the first three or only one and simple. In such a way we improve or sharpen previous results of Lazer and McKenna, Hofer, Gallouet and Kavian and of the author.

A note on a resonance problem

In this paper, we prove the existence of at least one solution to the problem where ∆ k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆ R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆ R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.

On the existence of a nontrivial solution to nonlinear problems at resonance

On considere le probleme −Δu=λ k u+g(u) dans Ω, u=0 sur δΩ, ou Ω est un domaine borne lisse dans R m et λ 0 <Λ 1 ≤λ 2 ≤… est la suite des valeurs propres de −Δ repetees autant de fois que leur multiplicite

On a class of dynamical systems with singular potential

We are concerned with the problem of searching for T-periodic solutions, T prescribed, for the equation -x=⊇V(x), where x(t) ∈ C 2 (R,R n ) and V ∈ C 1 (R/{0},R), in the case when V is «singular», i.e. V behaves like -k/|x|α near the origin

Preliminary communication: Nontrivial solutions of operator equations and morse indices of critical points of min-max type

On montre que si les hypotheses du theoreme de point selle sont satisfaites, si dim Y≥2, et si f −1 ({c}) contient seulement des points critiques non degeneres, alors f −1 ({c}) contient au moins un point critique u 0 tel que 2≤indice de Morse u 0 ≤dim Y