arXiv subjects

Riccardo Molle

Publications and source records attributed to Riccardo Molle.

22 records · Page 2Linked to original sources

arXiv2018

Multiple positive bound states for critical Schrödinger-Poisson systems

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system: $$ \left\{ \begin{array}{l} \vspace{2mm} -Δu+V(x)u+K(x)ϕ(x)u=u^5 -Δϕ=K(x)u^2\qquad x\in\R^3 \end{array}\right.\quad\quad (SP) $$ We remark that $(SP)$ exhibits a "double" lack of compactness because of the unboundedness of $\R^3$ and the critical growth of the nonlinear term and that in our assumptions ground state solutions of $(SP)$ do not exist.

math.AP↗

arXiv2013

Multiple Solutions for Scalar Field Equations with Potentials having "Subsidences"

In this paper the question of finding infinitely many solutions to the problem $-Δu+a(x)u=|u|^{p-2}u$, in $\mathbb{R}^N$, $u \in H^1(\mathbb{R}^N)$, is considered when $N\geq 2$, $p \in (2, 2N/(N-2))$, and the potential $a(x)$ is a positive function which is not required to enjoy symmetry properties. Assuming that $a(x)$ satisfies a suitable "slow decay at infinity" condition and, moreover, that its graph has some "dips", we prove that the problem admits either infinitely many nodal solutions either infinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.

math.AP↗

arXiv2013

Semilinear elliptic problems in unbounded domains with unbounded boundary

This paper deals with a class of singularly perturbed nonlinear elliptic problems $(P_\e)$ with subcritical nonlinearity. The coefficient of the linear part is assumed to concentrate in a point of the domain, as $\e\to 0$, and the domain is supposed to be unbounded and with unbounded boundary. Domains that enlarge at infinity, and whose boundary flattens or shrinks at infinity, are considered. It is proved that in such domains problem $(P_\e)$ has at least 2 solutions.

math.AP↗

arXiv2013

Positive solutions for a nonlinear elliptic problem with strong lack of compactness

This paper deals with the lack of compactness in nonlinear elliptic problems $(P)$. In particular, a domain $Ω$ is provided where not converging Palais-Smale sequences exist at every energy level. Nevertheless, it is proved that problem $(P)$ has infinitely many solutions on $Ω$.

math.AP↗