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In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem
when 
\[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \]
$(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if 
$N\geq 3$) and 
$(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if 
$N=2$), where 
$I_{\alpha }$ and 
$I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem
where 
\[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \]
$\alpha ,\,\beta \in (0,\,2)$ and 
$f,\,g$ have exponential critical growth in the Trudinger–Moser sense.
We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data have been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger convex means. Sending one of the Robin parameters to
$+\infty $
, we obtain mixed Robin/Dirichlet comparison results in shells. We prove similar results on balls and prove a comparison principle on generalized cylinders with mixed Robin/Neumann boundary conditions.
By assuming that the Kirchhoff term has $K$
degeneracy points and other appropriated conditions, we have proved the existence of at least $K$
positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have ordered $H_{0}^{1}(\Omega )$
-norms. A concentration phenomena of these solutions is verified as a parameter related to the area of a region under the graph of the reaction term goes to zero.
Let 
$c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all 
$k,l \in \{1, \ldots , d\};$ and 
$\Omega \subset \mathbb{R}^{d}$ be open with uniformly 
$C^{2}$ boundary. We consider the divergence form operator 
$A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in 
$L_p(\Omega )$ when the coefficient matrix satisfies 
$(C(x) \xi , \xi ) \in \Sigma _\theta$ for all 
$x \in \Omega$ and 
$\xi \in \mathbb{C}^{d}$, where 
$\Sigma _\theta$ be the sector with vertex 0 and semi-angle 
$\theta$ in the complex plane. We show that a sectorial estimate holds for 
$A_p$ for all 
$p$ in a suitable range. We then apply these estimates to prove that the closure of 
$-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.
This paper deals with the following fractional elliptic equation with critical exponent
where 
\[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\]
$\lambda$, 
$\bar {\nu }\in {{\mathfrak R}}$, 
$s\in (0,1)$, 
$2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$, 
$(-\Delta )^{s}$ is the fractional Laplace operator, 
$\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and 
$\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition 
$u=0$ in 
${{\mathfrak R}}^{N}\setminus \Omega$. Under suitable conditions on 
$\lambda$ and 
$\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as 
$|\bar \nu | \to \infty$ .
We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in 
$\mathbb {R}^{n}\times \mathbb {R}$
, 
$n\ge 2$
, of the form 
$(r,y(r))$
or 
$(r(y),y)$
, where 
$r=|x|$
, 
$x\in \mathbb {R}^{n}$
, is the radially symmetric coordinate and 
$y\in \mathbb {R}$
. More precisely, for any 
$\lambda>\frac {1}{n-1}$
and 
$\mu>0$
, we will give a new proof of the existence of a unique even solution 
$r(y)$
of the equation 
$\frac {r^{\prime \prime }(y)}{1+r^{\prime }(y)^{2}}=\frac {n-1}{r(y)}-\frac {1+r^{\prime }(y)^{2}}{\lambda (r(y)-yr^{\prime }(y))}$
in 
$\mathbb {R}$
which satisfies 
$r(0)=\mu $
, 
$r^{\prime }(0)=0$
and 
$r(y)>yr^{\prime }(y)>0$
for any 
$y\in \mathbb {R}$
. We will prove that 
$\lim _{y\to \infty }r(y)=\infty $
and 
$a_{1}:=\lim _{y\to \infty }r^{\prime }(y)$
exists with 
$0\le a_{1}<\infty $
. We will also give a new proof of the existence of a constant 
$y_{1}>0$
such that 
$r^{\prime \prime }(y_{1})=0$
, 
$r^{\prime \prime }(y)>0$
for any 
$0<y<y_{1}$
, and 
$r^{\prime \prime }(y)<0$
for any 
$y>y_{1}$
.
In this article, we study the observability (or equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the ‘degenerated directions’ of the subelliptic structure.
First, for any
$\gamma \geq 1$
, we establish a resolvent estimate for the Baouendi–Grushin-type operator
$\Delta _{\gamma }=\partial _x^2+\left \lvert x\right \rvert ^{2\gamma }\partial _y^2$
, which has step
$\gamma +1$
. We then derive consequences for the observability of the Schrödinger-type equation
$i\partial _tu-\left (-\Delta _{\gamma }\right )^{s}u=0$
, where
$s\in \mathbb N$
. We identify three different cases: depending on the value of the ratio
$(\gamma +1)/s$
, observability may hold in arbitrarily small time or only for sufficiently large times or may even fail for any time.
As a corollary of our resolvent estimate, we also obtain observability for heat-type equations
$\partial _tu+\left (-\Delta _{\gamma }\right )^su=0$
and establish a decay rate for the damped wave equation associated with
$\Delta _{\gamma }$
.
We consider the boundary Hardy–Hénon equation
where 
\[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \]
$B_1(0)\subset \mathbb {R}^{N}$
$(N\geq 3)$ is a ball of radial 
$1$ centred at 
$0$, 
$p>0$ and 
$\alpha \in \mathbb {R}$. We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case 
$0< p<({N+2})/({N-2})$, we establish the estimate of positive solutions. When 
$\alpha \leq -2$ and 
$p>1$, we give some conclusions with respect to nonexistence. When 
$\alpha >-2$ and 
$1< p<({N+2})/({N-2})$, we obtain the existence of positive solution for the corresponding Dirichlet problem. When 
$0< p\leq 1$ and 
$\alpha \leq -2$, we show the nonexistence of positive solutions. When 
$0< p<1$, 
$\alpha >-2$, we give some results with respect to existence and uniqueness of positive solutions.
We study the fractional parabolic Lichnerowicz equation
$$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$
where 
$p>0$
and
$ 0<s<1 $
. We establish a Liouville-type theorem for positive solutions in the case
$p>1$
and give a uniform lower bound of positive solutions when
$0<p\leq 1$
. In particular, when v is independent of the time variable, we obtain a similar result for the fractional elliptic Lichnerowicz equation
$$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$
with
$p>0$
and
$0<s<1$
. This extends the result of Brézis [‘Comments on two notes by L. Ma and X. Xu’, C. R. Math. Acad. Sci. Paris349(5–6) (2011), 269–271] to the fractional Laplacian.
We consider the fractional elliptic problem:
where B1 is the unit ball in ℝN, N ⩾ 3, s ∈ (0, 1) and p > (N + 2s)/(N − 2s). We prove that this problem has infinitely many solutions with slow decay O(|x|−2s/(p−1)) at infinity. In addition, for each s ∈ (0, 1) there exists Ps > (N + 2s)/(N − 2s), for any (N + 2s)/(N − 2s) < p < Ps, the above problem has a solution with fast decay O(|x|2s−N). This result is the extension of the work by Dávila, del Pino, Musso and Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) to the fractional case.
We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space
$\mathbb {R}^3$
, in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighborhood of the origin.
In this article, we prove the continuity of the horizontal gradient near a C1,Dini non-characteristic portion of the boundary for solutions to 
$\Gamma ^{0,{\rm Dini}}$ perturbations of horizontal Laplaceans as in (1.1) below, where the scalar term is in scaling critical Lorentz space L(Q, 1) with Q being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the 
$\Gamma ^{1,\alpha }$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.
