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We are concerned with the existence and asymptotic behavior of multiple radial sign-changing solutions with the nodal characterization for a Kirchhoff-type problem involving the nonlinearity $|u|^{p-2}u(2

We study concavity properties of positive solutions to the Logarithmic Schr\"odinger equation $-\Delta u=u\, \log u^2$ in a general convex domain with Dirichlet conditions. To this aim, we analyse the auxiliary Lane-Emden problems $-\Delta u = \sigma\, (u^q-u)$ and build, for any $\sigma>0$ and $q>1$, solutions $u_q$ such that $u_q^{(1-q)/2}$ is convex. By choosing $\sigma_q=2/(1-q)$ and letting $q \to 1^+$ we eventually construct a solution $u$ of the Logarithmic Schr\"odinger equation such that $\log u$ is concave. This seems one of the few attempts in studying concavity properties for superlinear, sign changing sources. To get the result, we both make inspections on the constant rank theorem and develop Liouville theorems on convex epigraphs, which might be useful in other frameworks.

In this paper we study convexity properties for quasilinear Lane-Emden-Fowler equations of the type $$ \begin{cases} -\Delta_p u = a(x) u^q & \quad \hbox{ in $\Omega$},\\ u >0 & \quad \hbox{ in $\Omega$}, \\ u =0 & \quad \hbox{ on $\partial \Omega$}, \end{cases} $$ when $\Omega \subset \mathbb{R}^N$ is a convex domain. In particular, in the subhomogeneous case $q \in [0,p-1]$, the solution $u$ inherits concavity properties from $a$ whenever assumed, while it is proved to be concave up to an error if $a$ is near to a constant. More general cases are also taken into account, including a wider class of nonlinearities. These results generalize some contained in [Kennington, Indiana Univ. Math. J., 1985] and [Sakaguchi, Ann. Sc. Norm. Super. Pisa, 1987]. Additionally, some results for the singular case $q \in [-1,0)$ and the superhomogeneous case $q>p-1$, $q \approx p-1$ are obtained. Some properties for the $p$-fractional Laplacian $(-\Delta)^s_p$, $s\in (0,1)$, $s \approx 1$, are shown as well. We highlight that some results are new even in the semilinear case $p=2$; in some of these cases, we deduce also uniqueness (and nondegeneracy) of the critical point of $u$.

In this paper, we are concerned with normalized solutions in $H_{r}^{1}(\mathbb{R}^{3}) \times H_{r}^{1}(\mathbb{R}^{3})$ for Hartree-Fock type systems with the form \be\lab{ Hartree-Fock} \left\{ \begin{array}{ll} -\Delta u +\alpha \phi _{u,v} u=\lambda _{1} u+\left | u \right | ^{2q-2} u+\beta \left | v \right | ^{q} \left | u \right | ^{q-2} u , \\ -\Delta v +\alpha \phi _{u,v} v=\lambda _{2} v+\left | v\right | ^{2q-2} v+\beta \left | u \right | ^{q} \left | v \right | ^{q-2} v , \\ \int_{\mathbb{R}^{3}}\left | u \right | ^{2} {\rm d}x=a_{1} , \quad \int_{\mathbb{R}^{3}}\left | v \right | ^{2} {\rm d}x=a_{2} , \nonumber\\ \end{array} where $$ \phi_{u, v}\left(x\right):=\int_{\mathbb{R}^{3}} \frac{u^{2}(y)+v^{2}(y)}{|x-y|} {\rm d}y \in D^{1,2}\left(\mathbb{R}^{3}\right). $$ Here $\alpha,\beta>0, a_1,a_2>0$ and $10$ when $10$ small when $\frac{4}{3}\le q < \frac{3}{2}$. The nonexistence of normalized solutions is also considered for $\frac{3}{2}\le q < \frac{5}{3}$. Also, the orbital stability of standing waves is obtained under local well-posedness assumptions of the evolution problem., Comment: 17 pages

We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schr\"{o}dinger systems \[ \left\{ \begin{array}{ll} \displaystyle -\Delta u +A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u)-a(x)|u|^{p-2}u, \newline \displaystyle \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,~\partial_1A_1+\partial_2A_2=0, \newline \displaystyle \partial_1A_0=A_2|u|^2,~ \partial_2A_0=-A_1|u|^2, \end{array} \right. \] where $a:\mathbb R^2\to\mathbb R^+$ is an external potential, $p\in(1,2)$ and $f\in \mathcal{C}(\mathbb R)$ denotes a nonlinearity that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. By introducing an improvement of the version of Trudinger-Moser inequality, we are able to investigate the existence of positive ground state solutions for the given system using variational method., Comment: 20 pages

We investigate the existence and concentration of normalized solutions for a $p$-Laplacian problem with logarithmic nonlinearity of type \[ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^p\Delta_p u+V(x)|u|^{p-2}u=\lambda |u|^{p-2}u+|u|^{p-2}u\log|u|^p ~\text{in}~\mathbb R^N,\newline \displaystyle \int_{\mathbb R^N}|u|^pdx=a^p\varepsilon^N, \end{array} \right. \] where $a,\varepsilon> 0$, $\lambda\in\mathbb R$ is known as the Lagrange multiplier, $\Delta_p\cdot =\text{div} (|\nabla \cdot|^{p-2}\nabla \cdot)$ denotes the usual $p$-Laplacian operator with $2\leq p < N$ and $V \in \mathcal{C}^0(\mathbb R^N)$ is the potential which satisfies some suitable assumptions. We prove that the number of positive solutions depends on the profile of $V$ and each solution concentrates around its corresponding global minimum point of $V$ in the semiclassical limit when $\varepsilon\to0^+$ using variational method. Moreover, we also get the existence of normalized solutions for some logarithmic $p$-Laplacian equations involving mass-supercritical nonlinearities., Comment: 35 pages

We are concerned with solutions of the following quasilinear Schr\"odinger equations \begin{eqnarray*} -{\mathrm{div}}\left(\varphi^{2}(u) \nabla u\right)+\varphi(u) \varphi^{\prime}(u)|\nabla u|^{2}+\lambda u=f(u), \quad x \in \mathbb{R}^{N} \end{eqnarray*} with prescribed mass $$ \int_{\mathbb{R}^{N}} u^{2} \mathrm{d}x=c, $$ where $N\ge 3, c>0$, $\lambda \in \mathbb{R}$ appears as the Lagrange multiplier and $\varphi\in C ^{1}(\mathbb{R} ,\mathbb{R}^{+})$. The nonlinearity $f \in C\left ( \mathbb{R}, \, \mathbb{R} \right )$ is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case., Comment: 18 pages, typos corrected

We study the fractional Schr\"{o}dinger equations coupled with a neutral scalar field $$ (-\Delta)^s u+V(x)u=K(x)\phi u +g(x)|u|^{q-2}u, \quad x\in \mathbb{R}^3,\qquad (I-\Delta)^t \phi=K(x)u^2, \quad x\in \mathbb{R}^3, $$ where $(-\Delta)^s$ and $(I-\Delta)^t$ denote the fractional Laplacian and Bessel operators with $\frac{3}{4}

In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb R}^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3} |u|^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t > 3, q\in(2,2^*_s), a>0$ and $\lambda,\mu>0$ parameters and $\alpha\in{\mathbb R}$ is an undetermined parameter. Under the $L^2$-subcritical perturbation $q\in (2, 2+\frac{4s}{3})$, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. For the $L^2$-supercritical perturbation $q\in (2+\frac{4s}{3}, 2^*_s)$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions., Comment: 43 pages

We study a class of planar Schr\"{o}dinger-Poisson systems $$ -\Delta u+\lambda V(x)u+\phi u=f(u) , \quad x\in{\mathbb R}^2,\qquad \Delta \phi=u^2, \quad x\in{\mathbb R}^2, $$ where $\lambda>0$ is a parameter, $V\in C({\mathbb R}^2,{\mathbb R}^+)$ has a potential well $\Omega \triangleq\text{int}\, V^{-1}(0)$ and the nonlinearity $f$ fulfills the supercritical exponential growth at infinity in the Trudinger-Moser sense. By exploiting the mountain-pass theorem and elliptic regular theory, we establish the existence and concentrating behavior of ground state solutions for sufficiently large $\lambda$., Comment: 33 pages

We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schr\"{o}dinger type problems with supercritical exponential growth $$ -\Delta u +\lambda u+A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u),\quad \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,\quad \partial_1A_1+\partial_2A_2=0,\quad \partial_1A_0=A_2|u|^2,\quad \partial_2A_0=-A_1|u|^2,\quad \int_{\mathbb{R}^2}|u|^2dx=a^2, $$ where $a\neq0$, $\lambda\in \mathbb{R}$ is known as the Lagrange multiplier and $f\in C^1(\mathbb{R})$ denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution., Comment: 39 pages

We get multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation $$ (-\Delta)^su+V(\varepsilon x)u=\lambda u+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where $(-\Delta)^s$ is the fractional Laplacian, $s\in(0,1)$, $a,\varepsilon>0$, $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:\mathbb{R}^N\rightarrow[0,+\infty)$ are bounded and continuous, and $f$ is continuous function with $L^2$-subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\varepsilon$ is small enough., Comment: 19 pages, revised version

We prove concavity properties for solutions to anisotropic quasi-linear equations, extending previous results known in the Euclidean case. We focus the attention on nonsmooth anisotropies and in particular we also allow the functions describing the anisotropies to be not even., Comment: 30 pages, revised version

In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion., Comment: 16 pages, types corrected, to appear on Asymptotic Analysis

In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schr\"odinger equations of the type $$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2} |\nabla u|^2 = f(u) \quad \hbox{in $\Omega$},$$ where $\Omega$ is a convex bounded domain of $\mathbb{R}^N$. In particular, we search for a function $\varphi:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $\varphi(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source., Comment: To be published on Differential and Integral Equations

In this paper, we establish some Stein-Weiss type inequalities with general kernels on the upper half space and study the existence of extremal functions for this inequality with the optimal constant. Furthermore, we also investigate the regularity, asymptotic estimates, symmetry and non-existence results of the positive solutions of the corresponding Euler-Lagrange integral system. As an application, we finally derive some Liouville type results for the Hartree type equations in the half space., Comment: 33 pages

In this paper, we are concerned with the coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} -\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3}+\beta v^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}\Delta v+b(x)v=\mu_{2}v^{3}+\beta u^{2}v \ \ \ \ \ \mbox{in}\ \mathbb{R}^{N}, \end{cases} \end{align*} where $1\leq N\leq3$, $\mu_{1},\mu_{2},\beta>0$, $a(x)$ and $b(x)$ are nonnegative continuous potentials, and $\varepsilon>0$ is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as $\varepsilon\rightarrow0$, when $a(x)$ and $b(x)$ achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary., Comment: 23 pages, typos corrected, to appear on Asymptotic Analysis

In this paper we are interested in the following critical Hartree equation \begin{equation*} \begin{cases} -\Delta u =\displaystyle{\Big(\int_{\Omega}\frac{u^{2_{\mu}^\ast} (\xi)}{|x-\xi|^{\mu}}d\xi\Big)u^{2_{\mu}^\ast-1}}+\varepsilon u ,~~~\text{in}~\Omega,\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}~\partial\Omega, \end{cases} \end{equation*} where $N\geq4$, $0<\mu\leq4$, $\varepsilon>0$ is a small parameter, $\Omega$ is a bounded domain in $\mathbb{R}^N$, and $2_{\mu}^\ast=\frac{2N-\mu}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. By establishing various versions of local Pohozaev identities and applying blow-up analysis, we first investigate the location of the blow-up points for single bubbling solutions to above the Hartree equation. Next we prove the local uniqueness of the blow-up solutions that concentrates at the non-degenerate critical point of the Robin function for $\varepsilon$ small., Comment: 40 pages

We are concerned with the study of existence of nontrivial ground states solutions for of Schr\"odinger systems with Chern-Simons gauge fields., Comment: 18 pages, typos corrected

We prove that quasi-concave positive solutions to a class of quasi-linear elliptic equations driven by the $p$-Laplacian in convex bounded domains of the plane have only one critical point. As a consequence, we obtain strict concavity results for suitable transformations of these solutions., Comment: 20 pages. To appear on a special issue of Rendiconti Lincei: Matematica ed Applicazioni dedicated to Antonio Ambrosetti

We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the $p$-Laplace operator and a general nonlinearity satisfying concavity type assumptions. This provides an extension of results previously known in the literature only for the torsion and the eigenfunction equations. In the semilinear case $p = 2$ the results are already new since they include new admissible nonlinearities., Comment: Example 2.1 modified, typos corrected. 20 pages

We study a class of gauged nonlinear Schr\"{o}dinger equations in the plane. We obtain existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, we prove existence of infinitely many solutions for odd nonlinearities., Comment: 22 pages, minor revision, accepted in DCDS-A

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity, in the theory of electrorheological fluids as well as in image processing for the regions where the variable exponent $p(x)$ reaches the value $1$., Comment: 15 pages

We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case., Comment: 18 pages

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are now-days available. The nonlinear case, corresponding to the $p$-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the \emph{nonlocal, nonlinear mean value kernel}, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) $p$-Laplacian (for $p\geq 2$) and to other gradient dependent nonlocal operators., Comment: 26 pages

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-\Delta)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $\Omega$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W^{s,p}_0(\Omega)$ and in the weighted H\"older space $C^0_s(\overline\Omega)$, respectively, do coincide., Comment: 14 pages

We obtain approximate convexity principles for solutions to some classes of nonlinear elliptic partial differential equations in convex domains involving approximately concave nonlinearities. Furthermore, we provide some applications to some meaningful special cases., Comment: 20 pages

We investigate the stability of ground states to a nonlinear focusing Schr\"odinger equation in presence of a Kirchhoff term. Through a spectral analysis of the linearized operator about ground states, we show a modulation stability estimate of ground states in the spirit of one due to Weinstein [{\it SIAM J. Math. Anal.}, 16(1985),472-491]., Comment: 11

We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $\Omega$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted H\"older regularity up to the boundary, that is, $u/d^s\in C^\alpha(\overline\Omega)$ for some $\alpha\in(0,1)$, $d$ being the distance from the boundary., Comment: 38 pages, 3 figures

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.

Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $\mathbb{R}^{*}\supset\mathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1707.09265

We provide new characterizations of Sobolev ad BV spaces in doubling and Poincare metric spaces in the spirit of the Bourgain-Brezis-Mironescu and Nguyen limit formulas holding in domains of R^N., Comment: 15 pages

We establish improved versions of the Hardy and Caffarelli-Kohn-Nirenberg inequalities by replacing the standard Dirichlet energy with some nonlocal nonconvex functionals which have been involved in estimates for the topological degree of continuous maps from a sphere into itself and characterizations of Sobolev spaces.

We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain-Brezis-Mironescu type limit formula for a related class of anisotropic nonlocal norms., Comment: 16 pages, typo corrections and improvements

In this paper, by using a compactness method, we study the Cauchy problem of the logarithmic Schr\"{o}dinger equation with harmonic potential. We then address the existence of ground states solutions as minimizers of the action on the Nehari manifold. Finally, we explicitly compute ground states (Gausson-type solution) and we show their orbital stability., Comment: 12 pages, updated version

By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth., Comment: 36 pages, revised version

We investigate two types of characterizations for anisotropic Sobolev and BV spaces. In particular, we establish anisotropic versions of the Bourgain-Brezis-Mironescu formula, including the magnetic case both for Sobolev and BV functions., Comment: 10 pages

In this paper we study the validity of a Gausson (soliton) dynamics of the logarithmic Schr\"odinger equation in presence of a smooth external potential., Comment: 18 pages

We prove that a nonlocal functional approximating the standard Dirichlet $p$-norm fails to decrease under two-point rearrangement. Furthermore, we get other properties related to this functional such as decay and compactness, and the Polya-Szeg\"o inequality for Riesz fractional gradients, a notion recently introduced in the literature., Comment: 12 pages

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends results in the literature for the N-Laplacian operator., Comment: 16 pages

We establish two new characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields in terms of nonlocal functionals. The first one is related to the BBM formula, due to Bourgain, Brezis, and Mironescu. The second one is related to the work of the first author on the classical Sobolev spaces. We also study the convergence almost everywhere and the convergence in $L^1$ appearing naturally in these contexts., Comment: 21 pages

We provide a new characterization of the logarithmic Sobolev inequality., Comment: 5 pages

By virtue of a suitable approximation argument, we prove a Pohozaev identity for nonlinear nonlocal problems on $\mathbb{R}^N$ involving the fractional $p-$Laplacian operator. Furthermore we provide an application of the identity to show that some relevant levels of the energy functional associated with the problem coincide., Comment: This paper has been withdrawn by the authors due to a crucial error in Proposition A.1