Concentration phenomena for a fractional Schr\"odinger-Kirchhoff type equation

In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schr\"odinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy + \frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2} dx\right)[\varepsilon^{2s} (-\Delta)^{s}u+ V(x)u]= f(u) \, \mbox{in} \mathbb{R}^{3} \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $(-\Delta)^{s}$ is the fractional Laplacian, $M$ is a Kirchhoff function, $V$ is a continuous positive potential and $f$ is a superlinear continuous function with subcritical growth. By using penalization techniques and Ljusternik-Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum.Comment: Mathematical Methods in the Applied Sciences (2017

Multiplicity and concentration results for some nonlinear Schr\"odinger equations with the fractional pp-Laplacian

We consider a class of parametric Schr\"odinger equations driven by the fractional $p$-Laplacian operator and involving continuous positive potentials and nonlinearities with subcritical or critical growth. By using variational methods and Ljusternik-Schnirelmann theory, we study the existence, multiplicity and concentration of positive solutions for small values of the parameter

A multiplicity result for a fractional Kirchhoff equation in RN\mathbb{R}^{N} with a general nonlinearity

In this paper we deal with the following fractional Kirchhoff equation \begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-\Delta)^{s}u = g(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $N\geq 2$, $p>0$, $q$ is a small positive parameter and $g: \mathbb{R}\rightarrow \mathbb{R}$ is an odd function satisfying Berestycki-Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that $q$ is sufficiently small

Partial Regularity Results for Asymptotic Quasiconvex Functionals with General Growth

We prove partial regularity for minimizers of vectorial integrals of the Calculus of Variations, with general growth condition, imposing quasiconvexity assumptions only in an asymptotic sense

Regularity results for asymptotic problems

Elliptic and parabolic equations arise in the mathematical description of a wide variety of phenomena, not only in the natural science but also in engineering and economics. To mention few examples, consider problems arising in different contexts: gas dynamics, biological models, the pricing of assets in economics, composite media. The importance of these equations from the applications' point of view is equally interesting from that of analysis, since it requires the design of novel techniques to attack the always valid question of existence, uniqueness and regularity of solutions. \\ In particular, in recent years parabolic problems came more and more into the focus of mathematicians. Changing from elliptic to the parabolic case means physically to switch from the stationary to the non-stationary case, i.e. the time is introduced as an additional variable. Exactly this natural origin constitutes our interest in parabolic problems: they reflect our perception of space and time. Therefore they often can be used to model physical process, e.g. heat conduction or diffusion process. \medskip \noindent In this thesis I will principally concentrate on the regularity properties of solutions of second order systems of partial differential equations in the elliptic and parabolic context. The outline of the thesis is as follows. \smallskip \noindent After giving some preliminary results, in the 3st Chapter we consider the parabolic analogue of some regularity results already known in the elliptic setting, concerning systems becoming parabolic only in an {\it asymptotic} sense. In the standard elliptic version, these results prove the {\it Lipschitz regularity} of solutions to elliptic systems of the type \dive a(Du)=0, with $u: \Omega \rightarrow \R^{N}$, under the main assumption that the vector field $a: \R^{Nn}\rightarrow \R^{Nn}$ is {\it asymptotically} close, in $C^{1}$-sense, to some regular vector field $b$. Therefore, one can ask what happens when the vector field $a$ is {\it asymptotically} close, in a $C^{0}$-sense, to the regular vector field $b(\xi)=\xi$. In this direction, in the parabolic framework, the first result obtained shows that the spatial gradient of $u$ belongs to $L^{\infty}_{\rm{loc}}$. \\ The question that naturally arises is what happens in case of power $p\neq 2$, and more in general in case of general growth \V. \noindent Regarding the general growth \V, in Chapter \ref{phi}, we study variational integrals of the type \begin{equation*} \mathcal{F}(u):=\int_{\Omega} f(Du) \,dx \quad \mbox{ for } u:\Omega \rightarrow \R^{N} \end{equation*} where $\Omega$ is an open bounded set in $\R^{n}$, $n\geq 2$, $N\geq 1$. Here $f:\R^{Nn}\rightarrow \R$ is a quasiconvex continuous function satisfying a non-standard growth condition: \begin{equation*} |f(z)|\leq C(1+ \varphi(|z|) ), \quad \forall z\in \R^{Nn}, \end{equation*} where $C$ is a positive constant and \V is a given $N$-function (see Section \ref{Orlicz} for more details about Orlicz functions). Exhibiting an adequate notion of strict W^{1,\V}-quasiconvexity at infinity, which we call W^{1,\V}-asymptotic quasiconvexity, we prove a partial regularity result, namely that minimizers are {\it Lipschitz} continuous on an open and dense subset of $\Omega$. \\ In the last Chapter we deal with the study of {\it local Lipschitz regularity} of weak solutions to non-linear second order parabolic systems of general growth \begin{equation}\label{Prob} u_{t}^{\beta} - \sum_{i=1}^{n} (\A_{i}^{\alpha}(Du))_{x_{i}}=0, \mbox{ in } \Omega_{T}:=\Omega \times (-T,0) \end{equation} where $\Omega$ is a bounded domain in $\R^{n}$, $n\geq 2$, $T>0$, $u:\Omega_{T} \rightarrow \R^{N}$, $N>1$ and \A is a tensor having general growth, that is \displaystyle{\A_{i}^{\alpha}(Du)= \frac{\V'(|Du|)}{|Du|} u_{x_{i}}^{\alpha}}, where \V is a given $N$-function. \\ Actually, having such result, as observed before, it is possible to prove the analogue of the first problem (studied in Chapter \ref{BMO}) in this case of nonstandard growth, considering an operator $\mathcal{A}$ that is {\it asymptotically} related to (\ref{Prob})

Existence, multiplicity and concentration for a class of fractional p&qp\&q Laplacian problems in RN\mathbb{R}^{N}

In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-\Delta)_{p}^{s}u+ (-\Delta)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(-\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $\varepsilon$ is sufficiently small.Comment: arXiv admin note: text overlap with arXiv:1709.0373

Infinitely many solutions for fractional Kirchhoff-Sobolev-Hardy critical problems

We investigate a class of critical stationary Kirchhoff fractional p-Laplacian problems in presence of a Hardy potential. By using a suitable version of the symmetric mountain-pass lemma due to Kajikiya, we obtain the existence of a sequence of infinitely many arbitrarily small solutions converging to zero

High-Resolution Conformational Analysis of RGDechi-Derived Peptides Based on a Combination of NMR Spectroscopy and MD Simulations

The crucial role of integrin in pathological processes such as tumor progression and metastasis formation has inspired intense efforts to design novel pharmaceutical agents modulating integrin functions in order to provide new tools for potential therapies. In the past decade, we have investigated the biological proprieties of the chimeric peptide RGDechi, containing a cyclic RGD motif linked to an echistatin C-terminal fragment, able to specifically recognize &alpha;v&beta;3 without cross reacting with &alpha;v&beta;5 and &alpha;IIb&beta;3 integrin. Additionally, we have demonstrated using two RGDechi-derived peptides, called RGDechi1-14 and &psi;RGDechi, that chemical modifications introduced in the C-terminal part of the peptide alter or abolish the binding to the &alpha;v&beta;3 integrin. Here, to shed light on the structural and dynamical determinants involved in the integrin recognition mechanism, we investigate the effects of the chemical modifications by exploring the conformational space sampled by RGDechi1-14 and &psi;RGDechi using an integrated natural-abundance NMR/MD approach. Our data demonstrate that the flexibility of the RGD-containing cycle is driven by the echistatin C-terminal region of the RGDechi peptide through a coupling mechanism between the N- and C-terminal regions

Evolving trends in the management of acute appendicitis during COVID-19 waves. The ACIE appy II study

Background: In 2020, ACIE Appy study showed that COVID-19 pandemic heavily affected the management of patients with acute appendicitis (AA) worldwide, with an increased rate of non-operative management (NOM) strategies and a trend toward open surgery due to concern of virus transmission by laparoscopy and controversial recommendations on this issue. The aim of this study was to survey again the same group of surgeons to assess if any difference in management attitudes of AA had occurred in the later stages of the outbreak. Methods: From August 15 to September 30, 2021, an online questionnaire was sent to all 709 participants of the ACIE Appy study. The questionnaire included questions on personal protective equipment (PPE), local policies and screening for SARS-CoV-2 infection, NOM, surgical approach and disease presentations in 2021. The results were compared with the results from the previous study. Results: A total of 476 answers were collected (response rate 67.1%). Screening policies were significatively improved with most patients screened regardless of symptoms (89.5% vs. 37.4%) with PCR and antigenic test as the preferred test (74.1% vs. 26.3%). More patients tested positive before surgery and commercial systems were the preferred ones to filter smoke plumes during laparoscopy. Laparoscopic appendicectomy was the first option in the treatment of AA, with a declined use of NOM. Conclusion: Management of AA has improved in the last waves of pandemic. Increased evidence regarding SARS-COV-2 infection along with a timely healthcare systems response has been translated into tailored attitudes and a better care for patients with AA worldwide