Giovanna Cerami

Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents

Abstract In this paper we study the existence of nontrivial solutions for the boundary value problem { − Δ u − λ u − u | u | 2 ⁎ − 2 = 0 in Ω u = 0 on ∂ Ω when Ω⊂Rn is a bounded domain, n ⩾ 3, 2 ⁎ = 2 n ( n − 2 ) is the critical exponent for the Sobolev embedding H 0 1 ( Ω ) ⊂ L p ( Ω ) , λ is a real parameter. We prove that there is bifurcation from any eigenvalue λj of − Δ and we give an estimate of the left neighbourhoods ] λ j ⁎ , λj] of λj, j∈N, in which the bifurcation branch can be extended. Moreover we prove that, if λ ∈ ] λ j ⁎ , λj[, the number of nontrivial solutions is at least twice the multiplicity of λj. The same kind of results holds also when Ω is a compact Riemannian manifold of dimension n ⩾ 3, without boundary and Δ is the relative Laplace-Beltrami operator.

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.

Existence and Multiplicity Results for Some Scalar Fields Equations

In this paper the results of some researches concerning Scalar Field Equations are summarized. The interest is focused on the question of existence and multiplicity of stationary solutions, so the model equation \( -\Delta u + a(x)u = |u|^{p-1}u \; \; \rm{in} \; \mathbb{R}^{N} \) is considered. The difficulties and the ideas introduced to face them as well as some well known results are discussed. Some recent advances concerning existence and multiplicity of multi-bump solutions are described in more detail.

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.