Related papers: Monotonicity formulas for coupled elliptic gradien…
This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schr\"odinger) equation $(\Delta + k^2 q) u = 0$ in a bounded domain for fixed non-resonance frequency $k>0$ and real-valued…
We study the system $-\Delta \mathbf{u}=| \mathbf{u}|^{\alpha-1} \mathbf{u}$ with $1<\alpha\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a…
We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ \Delta^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index…
We consider the following elliptic system with fractional laplacian $$ -(-\Delta)^su=uv^2,\ \ -(-\Delta)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n,$$ where $s\in(0,1)$ and $(-\Delta)^s$ is the $s$-Lapalcian. We first prove that all positive…
By applying a high-dimensional parabolic-to-elliptic transformation, we establish a monotonicity formula for the extension problem of the fractional parabolic semilinear equation $(\partial_t -\Delta)^s u = |u|^{p-1}u$, where $0<s<1$. This…
We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…
We develop a monotonicity formula for solutions of the fractional Toda system $$ (-\Delta)^s f_\alpha = e^{-(f_{\alpha+1}-f_\alpha)} - e^{-(f_\alpha-f_{\alpha-1})} \quad \text{in} \ \ \mathbb R^n,$$ when $0<s<1$, $\alpha=1,\cdots,Q$,…
We consider the equation \[ -\Delta u = |x|^{\alpha} |u|^{p-1}u, \ \ x \in B, \ \ u=0 \quad \text{on} \ \ \partial B, \] where $B \subset {\mathbb R}^2$ is the unit ball centered at the origin, $\alpha \geq0$, $p>1$, and we prove some…
We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a H\"older continuous linear term. With the help of those formulas we are able to…
The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: \begin{align*} \begin{cases} -div(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ u &=0 \quad…
We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…
This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$,…
In this paper, we are concerned with the monotonic and symmetric properties of convex solutions to fully nonlinear elliptic systems. We mainly discuss Monge-Amp\`ere type systems for instance, considering \begin{equation*}…
We study the symmetry properties for solutions of elliptic systems of the type {ll}-\dive(a_1(x,|\nabla u^1|(X))\nabla u^1(X))=F_{1}(x, u^1(X),..., u^n(X)), ... -\dive(a_n(x,|\nabla u^n|(X))\nabla u^n(X))=F_{n}(x, u^1(X),..., u^n(X)), where…
In this paper we prove the monotonicity of positive solutions to $ -\Delta_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$.…
We study the asymptotic behavior of positive radial solutions for quasilinear elliptic systems that have the form \begin{equation*} \left\{ \begin{aligned} \Delta_p u &= c_1|x|^{m_1} \cdot g_1(v) \cdot |\nabla u|^{\alpha} &\quad\mbox{ in }…
We study a singular elliptic problem driven by a mixed local-nonlocal operator of the form \begin{equation*} \begin{aligned} -\Delta_p u + (-\Delta_q)^s u &= \frac{\lambda}{u^{\delta}} + u^r \text{ in } \Omega\newline u > 0 \text{ in }…
Using a physically motivated stress energy tensor, we prove weak and strong monotonicity formulas for solutions to the semilinear elliptic system $\Delta u=\nabla W(u)$ with $W$ nonnegative. In particular, we extend a recent two dimensional…
In this paper, we study the existence of nontrivial solutions of the Dirichlet boundary value problem for the following elliptic system: \begin{equation} \left\{ \begin{aligned} -\Delta u & = au + bv + f(x,u,v); &\quad\mbox{ for…
All correlation measures, classical and quantum, must be monotonic under local operations. In this paper, we characterize monotonic formulas that are linear combinations of the von Neumann entropies associated with the quantum state of a…