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Giusi Vaira

Bio: Giusi Vaira is an academic researcher from Seconda Università degli Studi di Napoli. The author has contributed to research in topics: Riemannian manifold & Boundary (topology). The author has an hindex of 14, co-authored 45 publications receiving 909 citations. Previous affiliations of Giusi Vaira include Sapienza University of Rome & International School for Advanced Studies.

Papers
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Journal ArticleDOI
TL;DR: In this article, the existence of positive solutions for the Schrodinger-Poisson system with nonnegative functions has been proved, but not requiring any symmetry property on them and satisfying suitable assumptions.

306 citations

Journal ArticleDOI
TL;DR: In this paper, 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 was studied.
Abstract: 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.

129 citations

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TL;DR: In this paper, the existence of positive ground states with minimal energy was shown in the case of an elliptic system and satisfying suitable assumptions, but not requiring any symmetry property on them.
Abstract: In this paper we consider the following elliptic system in $${\mathbb{R}^3}$$ $$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.$$ where λ is a real parameter, $${p\in (1, 5)}$$ if λ < 0 while $${p\in (3, 5)}$$ if λ > 0 and K(x), a(x) are non-negative real functions defined on $${\mathbb{R}^3}$$ . Assuming that $${\lim_{|x|\rightarrow+\infty}K(x)=K_{\infty} >0 }$$ and $${\lim_{|x|\rightarrow+\infty}a(x)=a_{\infty} >0 }$$ and satisfying suitable assumptions, but not requiring any symmetry property on them, we prove the existence of positive ground states, namely the existence of positive solutions with minimal energy.

72 citations

Posted Content
TL;DR: In this article, the authors considered the problem of solving a problem in the system in the form of a problem-madipartenza-0 problem, where the problem is solved in the following way:
Abstract: In this paper we consider the system in $\R^3$ \label{problemadipartenza0} -\e^2\Delta u+V(x)u+\phi(x)u=u^{p},

68 citations

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TL;DR: In this paper, the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential $V(x) was proved. But such solutions do not exist in the framework of the usual Nonlinear Schrodinger Equation.
Abstract: In this paper we consider the system in $\mathbb{R}^3$ \begin{equation} \left\{ \begin{array}{l} -\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p, \\ -\Delta \phi = u^2, \end{array} \right. \end{equation} for $p\in (1,5)$. We prove the existence of multi-bump solutions whose bumps concentrate around a local minimum of the potential $V(x)$. We point out that such solutions do not exist in the framework of the usual Nonlinear Schrodinger Equation.

59 citations


Cited by
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Journal ArticleDOI
TL;DR: A survey of the existence and properties of solutions to the Choquard type equations can be found in this paper, where some variants and extensions of its variants can also be found.
Abstract: We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$\begin{aligned} -\Delta u + V(x)u = \left( |x|^{-(N-\alpha )} *|u |^p\right) |u |^{p - 2} u \quad \text {in} \ \mathbb {R}^N, \end{aligned}$$ and some of its variants and extensions.

352 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of bound states of the nonlinear Schrodinger-Poisson system has been studied in the context of critical point theory and perturbation methods.
Abstract: We discuss some recent results dealing with the existence of bound states of the nonlinear Schrodinger-Poisson system $$\left\{ \begin{gathered} - \Delta u + V(x)u + \lambda K(x)\phi (x)u = |u|^{{p - 1}} u, \hfill \\ - \Delta \phi = K(x)u^{2}, \hfill \\ \end{gathered} \right.$$ as well as of the corresponding semiclassical limits. The proofs are based upon Critical Point theory and Perturbation Methods.

216 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of sign-changing solutions for the Schrodinger-Poisson system was investigated and invariant sets of descending flow invariants were used to prove that the system has infinitely many sign changing solutions.
Abstract: In this paper, we consider the following Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=f(u)&{}\quad \text{ in }\ \mathbb {R}^3,\\ -\Delta \phi =u^2&{}\quad \text{ in }\ \mathbb {R}^3. \end{array} \right. \end{aligned}$$ We investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. In particular, the nonlinear term includes the power-type nonlinearity $$f(u)=|u|^{p-2}u$$ for the well-studied case $$p\in (4,6)$$ , and the less studied case $$p\in (3,4)$$ , and for the latter case, few existence results are available in the literature.

170 citations

Journal ArticleDOI
TL;DR: A survey of the existence and properties of solutions to the Choquard type equations can be found in this article, where some variants and extensions of its variants can also be found.
Abstract: We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$ -\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$}, $$ and some of its variants and extensions.

164 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of nontrivial solution and concentration results are obtained via variational methods under suitable assumptions on V and K. In particular, the potential V is allowed to be sign-changing for the case p ∈ ( 4, 6 ).

161 citations