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In this paper we prove some results on the boundary behavior of solutions to fractional elliptic problems. Firstly, we establish a Hopf Lemma for solutions to some integro-differential equations. The main novelty of our result is that we do not assume any global condition on the sign of the solutions. Secondly, we show that non-trivial radial solutions cannot have infinitely many zeros accumulating at the boundary. We provide concrete examples to show that the results obtained are sharp.

We investigate existence and qualitative properties of globally defined and positive radial solutions of the Lane-Emden system, posed on a Cartan-Hadamard model manifold $ \mathbb{M}^n $. We prove that, for critical or supercritical exponents, there exists at least a one-parameter family of such solutions. Depending on the stochastic completeness or incompleteness of $ \mathbb{M}^n $, we show that the existence region stays one dimensional in the former case, whereas it becomes two dimensional in the latter. Then, we study the asymptotics at infinity of solutions, which again exhibit a dichotomous behavior between the stochastically complete (where both components are forced to vanish) and incomplete cases. Finally, we prove a rigidity result for finite-energy solutions, showing that they exist if and only if $ \mathbb{M}^n $ is isometric to $ \mathbb{R}^n $.

In this paper we are concerned with the existence of normalized solutions for nonlinear Schr\"odinger equations on noncompact metric graphs with localized nonlinearities. In a $L^2$-supercritical regime, we obtain the existence of solutions for any prescribed mass. This result is obtained through an approach which could prove successful to treat more general equations on noncompact graphs., Comment: arXiv admin note: text overlap with arXiv:2204.01043

In this paper, we study, for functionals having a mountain pass geometry on a constraint, the existence of bounded Palais-Smale sequences carrying Morse index type information.

We investigate the structure of the nodal set of solutions to \[ -\Delta u = \lambda_+(u^+)^{p-1}-\lambda_-(u^-)^{q-1} \] where $1 \le p0$, $\lambda_- \ge 0$. The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.

This paper is devoted to the existence of non-trivial bound states of prescribed mass for the mass-supercritical nonlinear Schr\"odinger equation on compact metric graphs. The investigation is based upon a general variational principle which combines the monotonicity trick and a min-max theorem with second order information, and upon the blow-up analysis of bound states with prescribed mass and bounded Morse index.

The Cartan-Hadamard conjecture states that, on every $n$-dimensional Cartan-Hadamard manifold $ \mathbb{M}^n $, the isoperimetric inequality holds with Euclidean optimal constant, and any set attaining equality is necessarily isometric to a Euclidean ball. This conjecture was settled, with positive answer, for $n \le 4$. It was also shown that its validity in dimension $n$ ensures that every $p$-Sobolev inequality ($ 1 < p < n $) holds on $ \mathbb{M}^n $ with Euclidean optimal constant. In this paper we address the problem of classifying all Cartan-Hadamard manifolds supporting an optimal function for the Sobolev inequality. We prove that, under the validity of the $n$-dimensional Cartan-Hadamard conjecture, the only such manifold is $ \mathbb{R}^n $, and therefore any optimizer is an Aubin-Talenti profile (up to isometries). In particular, this is the case in dimension $n \le 4$. Optimal functions for the Sobolev inequality are weak solutions to the critical $p$-Laplace equation. Thus, in the second part of the paper, we address the classification of radial solutions (not necessarily optimizers) to such a PDE. Actually, we consider the more general critical or supercritical equation \[ -��_p u = u^q \, , \quad u>0 \, , \qquad \text{on } \mathbb{M}^n \, , \] where $q \ge p^*-1$. We show that if there exists a radial finite-energy solution, then $\mathbb{M}^n$ is necessarily isometric to $\mathbb{R}^n$, $q=p^*-1$ and $u$ is an Aubin-Talenti profile. Furthermore, on model manifolds, we describe the asymptotic behavior of radial solutions not lying in the energy space $\dot{W}^{1,p}(\mathbb{M}^n)$, studying separately the $p$-stochastically complete and incomplete cases., Accepted for publication on Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

We prove an Alt-Caffarelli-Friedman montonicity formula for pairs of functions solving elliptic equations driven by different ellipticity matrices in their positivity sets. As application, we derive Liouville-type theorems for subsolutions of some elliptic systems, and we analyze segregation phenomena for systems of equations where the diffusion of each density is described by a different operator.

We consider the following critical weakly coupled elliptic system \[ \begin{cases} -\Delta u_i = \mu_i |u_i|^{2^*-2}u_i + \sum_{j \neq i} \beta_{ij} |u_j|^{\frac{2^*}{2}} |u_i|^{\frac{2^*-4}{2}} u_i & \text{in $\Omega_\varepsilon$} u_i >0 & \text{in $\Omega_\varepsilon$} u_i = 0 & \text{on $\partial \Omega_\varepsilon$},\end{cases} \qquad i =1,\dots,m, \] in a domain $\Omega_\varepsilon \subset \mathbb{R}^N$, $N=3,4$, with small shrinking holes as the parameter $\varepsilon \to 0$. We prove the existence of positive solutions of two different types: either each density concentrates around a different hole, or we have groups of components such that all the components within a single group concentrate around the same point, and different groups concentrate around different points.

We prove the existence of infinitely many solutions $��_1, ��_2 \in \mathbb{R}$, $u,v \in H^1(\mathbb{R}^3)$, for the nonlinear Schr��dinger system \[ \begin{cases} -��u - ��_1 u = ��u^3+ ��u v^2 & \text{in $\mathbb{R}^3$} -��v- ��_2 v = ��v^3 +��u^2 v & \text{in $\mathbb{R}^3$} u,v>0 & \text{in $\mathbb{R}^3$} \int_{\mathbb{R}^3} u^2 = a^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a^2, \end{cases} \] where $a,��>0$ and $��\le -��$ are prescribed. Our solutions satisfy $u\ne v$ so they do not come from a scalar equation., 21 pages, substantial difference with respect to the previous version, the section 3 is entirely new

We consider the system of coupled elliptic equations \[ \begin{cases} -��u - ��_1 u = ��_1 u^3+ ��u v^2 \\ -��v- ��_2 v = ��_2 v^3 +��u^2 v \end{cases} \text{in $\mathbb{R}^3$}, \] and study the existence of positive solutions satisfying the additional condition \[ \int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2. \] Assuming that $a_1,a_2,��_1,��_2$ are positive fixed quantities, we prove existence results for different ranges of the coupling parameter $��>0$. The extension to systems with an arbitrary number of components is discussed, as well as the orbital stability of the corresponding standing waves for the related Schr��dinger systems.

We study regularity issues for systems of elliptic equations of the type \[ -��u_i=f_{i,��}(x)-��\sum_{j\neq i} a_{ij} u_i |u_i|^{p-1}|u_j|^{p+1} \] set in domains $��\subset \mathbb{R}^N$, for $N \geq 1$. The paper is devoted to the derivation of $\mathcal{C}^{0,��}$ estimates that are uniform in the competition parameter $��> 0$, as well as to the regularity of the limiting free-boundary problem obtained for $��\to + \infty$. The main novelty of the problem under consideration resides in the non-trivial grouping of the densities: in particular, we assume that the interaction parameters $a_{ij}$ are only non-negative, and thus may vanish for specific couples $(i,j)$. As a main consequence, in the limit $��\to +\infty$, densities do not segregate pairwise in general, but are grouped in classes which, in turn, form a mutually disjoint partition. Moreover, with respect to the literature, we consider more general forcing terms, sign-changing solutions, and an arbitrary $p>0$. In addition, we present a regularity theory of the emerging free-boundary, defined by the interface among different segregated groups. These equations are very common in the study of Bose-Einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues., Final version, Nonlinear Analysis: Theory, Methods & Applications, 138 (2016), 388--427. Special Volume in honor of Juan Luis V\'azquez for his 70th birthday