Browsing by Author "Salvatore, Addolorata"

Now showing 1 - 2 of 2

Access status: Open Access ,
Infinitely many solutions for some nonlinear supercritical problems with break of symmetry
(Wydawnictwa AGH, 2019) Candela, Anna Maria; Salvatore, Addolorata
In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem $\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x)&\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}$ where $\Omega \subset \mathbb{R}^N$ is an open bounded domain, $N\geq 3$, and $A(x,t,\xi)$, $g(x,t)$, $h(x)$ are given functions, with $AA_t = \frac{\partial A}{\partial t}$, $a = \nabla_{\xi} A$, such that $A(x,\cdot,\cdot)$ is even and $g(x,\cdot)$ is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if $A(x,t,\xi)$ grows fast enough with respect to $t$, then the nonlinear term related to $g(x,t)$ may have also a supercritical growth.
Access status: Open Access ,
Some multiplicity results of homoclinic solutions for second order Hamiltonian systems
(Wydawnictwa AGH, 2020) Barile, Sara; Salvatore, Addolorata
We look for homoclinic solutions $q:\mathbb{R} \rightarrow \mathbb{R}^N$ to the class of second order Hamiltonian systems $-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}$ where $L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}$ and $a,b: \mathbb{R}\rightarrow \mathbb{R}$ are positive bounded functions, $G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}$ are positive homogeneous functions and $f:\mathbb{R}\rightarrow\mathbb{R}^N$. Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if $f\equiv 0$ and the existence of at least three solutions if $f$ is not trivial but small enough.