Giusi Vaira

On Concentration of Positive Bound States for the Schrödinger-Poisson Problem with Potentials

Abstract We study the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration.

SOLUTIONS OF THE SCHRÖDINGER–POISSON PROBLEM CONCENTRATING ON SPHERES, PART I: NECESSARY CONDITIONS

In this paper we study a coupled nonlinear Schrodinger–Poisson problem with radial functions. This system has been introduced as a model describing standing waves for the nonlinear Schrodinger equations in the presence of the electrostatic field. We provide necessary conditions for concentration on sphere for the solutions of this kind of problem extending the results already known.

Steady states with unbounded mass of the Keller–Segel system

We consider the stationary Keller-Segel system from chemotaxis in a ball and we show the existence of a solution concentrating at the boundary of the ball.

Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

AbstractLet (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ0 with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem (P𝜖)−ℒgu+𝜖u=uN+2N−2in(M,g).

Existence of bound states for schrödinger-newton type systems

(P_{\epsilon })\qquad -\mathcal {L}_{g} u+\epsilon u=u^{\frac {N+2}{N-2}}\ \text { in }\ (M,g) .

Infinitely many positive solutions for a Schrödinger–Poisson system ☆

We prove that for any k ∈ ℕ, there exists εk > 0 such that for all ε ∈ (0, εk) the problem (P𝜖) has a symmetric solution uε, which looks like the superposition of k positive bubbles centered at the point ξ0 as ε → 0. In particular, ξ0 is a towering blow-up point.

Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities

Abstract In this paper we consider the following elliptic system in ℝ3 where K(x), α(x) are non-negative real functions defined on ℝ3 so that . When K(x) ≡ K∞ and α(x) ≡ α∞ we have already proved the existence of a radial ground state of the above system. Here, by using a new version of the moving plane method, we show that all positive solutions of the above system with K(x) ≡ K∞ and a(x) ≡ α∞ are radially symmetric and the linearized operator around a radial ground state is also non-degenerate. Using these results we further prove, under additional assumptions on K(x) and α(x), but not requiring any symmetry property on them, the existence of a positive solution for the system.

Concentration along Geodesics for a Nonlinear Steklov Problem Arising in Corrosion Modeling

Abstract We are interested in the existence of infinitely many positive non-radial solutions of a Schrodinger–Poisson system with a positive radial bounded external potential decaying at infinity.

Infinitely many positive solutions for a Schrödinger-Poisson system

We consider equations of the form

Non-radial sign-changing solutions for the Schrödinger–Poisson problem in the semiclassical limit

\Delta u +\lambda^2 V(x)e^{\,u}=\rho