Riccardo Molle

Multiple positive solutions for singularly perturbed elliptic problems in exterior domains

Abstract The equation −e2Δu+ae(x)u=up−1 with boundary Dirichlet zero data is considered in an exterior domain Ω = R N ⧹ ω (ω bounded and N⩾2). Under the assumption that ae⩾a0>0 concentrates round a point of Ω as e→0, that p>2 and p

Nonlinear elliptic equations with critical Sobolev exponent in nearly starshaped domains

Under suitable assumptions on Ω, we show that, for e>0 small and k large enough, problem (1) below has solutions which concentrate and blow-up as e→0 at exactly k points; the blowing-up points approach ∂Ω as k→∞; the number of solutions tends to infinity as e→0. These assumptions allow Ω to be contractible and even arbitrarily close to starshaped domains. To cite this article: R. Molle, D. Passaseo, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 1029–1032.

On the Existence of Positive Solutions of Slightly Supercritical Elliptic Equations

Abstract In this paper we deal with existence and multiplicity of solutions for problem P(ε, Ω) below, in bounded domains Ω which do not satisfy some symmetry prop erties that play an important role in the proof of previous results on this subject. For ε > 0 small and k large enough, we construct solutions blowing up, as ε → 0, at exactly k points which approach the boundary of Ω as k tends to infinity. We also present some examples showing that solutions of this type also exist in some contractible domains which may be even very close to starshaped domains.

A finite-dimensional reduction method for slightly supercritical elliptic problems.

We describe a finite-dimensional reduction method to find solutions for a class of slightly supercritical elliptic problems. A suitable truncation argument allows us to work in the usual Sobolev space even in the presence of supercritical nonlinearities: we modify the supercritical term in such a way to have subcritical approximating problems; for these problems, the finite-dimensional reduction can be obtained applying the methods already developed in the subcritical case; finally, we show that, if the truncation is realized at a sufficiently large level, then the solutions of the approximating problems, given by these methods, also solve the supercritical problems when the parameter is small enough.

Positive bound state solutions for some Schrödinger–Poisson systems

The paper deals with a class of Schrodinger–Poisson systems, where the coupling term and the other coefficients do not have any symmetry property. Moreover, the setting we consider does not allow the existence of ground state solutions. Under suitable assumptions on the decay rate of the coefficients, we prove existence of a bound state, finite energy solution.

On the first curve of the Fučík spectrum for elliptic operators

In this Note we present a new variational characterization of the first nontrivial curve of the Fucik spectrum for elliptic operators with Dirichlet or Neumann boundary conditions. Moreover, we describe the asymptotic behaviour and some properties of this curve and of the corresponding eigenfunctions. In particular, this new characterization allows us to compare the first curve of the Fucik spectrum with the infinitely many curves we obtained in previous works: for example, we show that these curves are all asymptotic to the same lines as the first curve, but they are all distinct from such a curve

An elliptic problem with pointwise constraint on the laplacian

This paper deals with a class of variational inequalities, coming from variational problems with unilateral constraints. The presence of the constraint modifies the structure of the corresponding functional and increases the topological complexity of its sublevels, giving rise to some phenomena which are typical of nonlinear elliptic equations. Let Ω be a bounded domain of R, λ a real parameter, ψ and h two functions in H 0 (Ω) and in L (Ω) respectively. Set Kψ = {u ∈ H 0 (Ω) | ∆u ≤ ∆ψ (in weak sense)} (Kψ is a convex cone with vertex at ψ) and consider the problem

Multispike Solutions of Nonlinear Elliptic Equations with Critical Sobolev Exponent

This paper deals with existence and multiplicity of positive solutions for the equation , ϵ > 0, with zero Dirichlet boundary condition in bounded domains. For large k and small ϵ, the existence of k-spike solutions is proved under suitable assumptions on the shape of the domain. In particular, domains that are arbitrarily close to starshaped domains are allowed (no positive solution there exists if the domain is starshaped).

Multiple positive solutions for nonautonomous quasicritical elliptic problems in unbounded domains

Abstract The problem with boundary Dirichlet zero data is considered in an exterior domain Ω ⊂ ℝN. Assuming that, as ε →0, aε concentrates and blows up at a point of Ω, namely the existence of at least 2 distinct positive solutions is proved, if is suitably small. Furthermore, if aε(x) has a suitable behaviour at infinity, the existence of another positive solution is shown.

A Multiplicity Result for Singularly Perturbed Problems in Topologically Nontrivial Domains

Abstract The problem − ε2Δu + aε(x)u = up−1 with zero Dirichlet boundary condition is considered in a nontrivial bounded domain Ω ⊂ ℝN. Under the assumption that aε(x) ≥ a0 > 0 concentrates at a point of Ω as ε → 0 and has a suitable behaviour at infinity and, moreover, that p > 2 and if N ≥ 3, the existence of at least (cat Ω̅) + 2 distinct positive solutions is proved.