Multiparameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with convection term

Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations

We consider a class of weighted Emden-Fowler equations \begin{equation} \tag{$\mathcal P_α$} \label{eqab} \left\{\begin{array}{ll} -Δu=V_α (x) \, u^p & \text{in} \,\,B,\\ u>0 & \text{in} \,\,B,\\ u=0 & \text{on}\,\,\partial B, \end{array}\right. \end{equation} posed on the unit ball $B=B(0,1)\subset \mathbb R^N$, $N \geq1$. We prove that symmetry breaking occurs for the groundstate solutions as the parameter $α\rightarrow \infty.$ The above problem reads as a possibly large perturbation of the classical Hénon equation. We consider a radial function $V_α$ having a spherical shell of zeroes at $…

Blow up at finite time for wave equation in visco-elasticity: a new kind for one spatial variable Emden-Fowler type

For one spatial variable, a new kind of nonlinear wave equation for Emden-Fowler type is considered with boundary value null and initial values. Under certain conditions on the initial data and the exponent p, we exhibit that the visco-elastic term leads our problem to be dissipative and the global solutions still non-exist in L2 at given finite time.

变分联系Olech-Opial不等式、Wirtinger不等式与Emden-Fowler方程

A Variational link between the Olech-Opial inequality, the Wirtinger inequality, and Emden-Fowler equations

本文通过混合能量项的积分恒等式，建立Olech-Opial不等式与Wirtinger不等式之间的变分联系，并通过变分问题得到最优常数，其极值满足Emden-Fowler方程。

Entire positive solutions of the singular Emden-Fowler equation with nonlinear gradient term

Let $p$ and $q$ be locally Hölder functions in $\RR^N$, $p>0$ and $q\geq 0$. We study the Emden-Fowler equation $-Δu+ q(x)|\nabla u|^a=p(x)u^{-γ}$ in $\RR^N$, where $a$ and $γ$ are positive numbers. Our main result establishes that the above equation has a unique positive solutions decaying to zero at infinity. Our proof is elementary and it combines the maximum principle for elliptic equations with a theorem of Crandall, Rabinowitz and Tartar.

The Emden-Fowler equation on a spherical cap of $\mathbb{S}^N$

Let $\mathbb{S}^N\subset\mathbb{R}^{N+1}$, $N\ge 3$, be the unit sphere, and let $S_Θ\subset\mathbb{S}^N$ be a geodesic ball with geodesic radius $Θ\in(0,π)$. We study the bifurcation diagram $\{(Θ,\left\|U\right\|_{\infty})\}\subset\mathbb{R}^2$ of the radial solutions of the Emden-Fowler equation on $S_Θ$ $Δ_{\mathbb{S}^N}U+U^p=0$ in $S_Θ$, $U=0$ on $\partial S_Θ$, $U>0$ in $S_Θ$, where $p>1$. Among other things, we prove the following: For each $p>p_{\rm S}:=(N-2)/(N+2)$, there exists $\underlineΘ\in(0,π)$ such that the problem has a radial solution for $Θ\in(\underlineΘ,π)$ and has no radi…

有界域上耦合子线性兰-埃门德-福勒系统的适定性与单调分析

Well-Posedness and Monotone Analysis for a Coupled Sublinear Lane--Emden--Fowler System on Bounded Domains

本文研究了有界域上耦合子线性兰-埃门德-福勒系统的适定性与单调分析，通过次解和上解方法及单调迭代方案，证明了该系统存在唯一的正解，并展示了解对数据的连续依赖性。

Lipschitz领域中奇异Lane-Emden-Fowler方程的边界正则性理论

Boundary regularity theory of the singular Lane-Emden-Fowler equation in a Lipschitz domain

研究奇异Lane-Emden-Fowler方程的适定性及边界处解的增长率，通过分类边界点的极限锥频率，获得不同情况下的增长率估计，并推导出奇异Lane-Emden-Fowler方程的边界Harnack原理，为非光滑领域中奇异椭圆方程的行为提供新见解。

A new dynamical approach of Emden-Fowler equations and systems

We give a new approach on general systems of the form \[(G){[c]{c}% -Δ_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=ε_{1}|x| ^{a}u^{s}v^δ, -Δ_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla u)=ε_{2}|x|^{b}u^μv^{m},\] where $Q,p,q,δ,μ,s,m,$ $a,b$ are real parameters, $Q,p,q\neq1,$ and $ε_{1}=\pm1,$ $ε_{2}=\pm1.$ In the radial case we reduce the problem to a quadratic system of order 4, of Kolmogorov type. Then we obtain new local and global existence or nonexistence results. In the case $ε_{1}=ε_{2}=1,$ we also describe the behaviour of the ground states in two cases where the system is va…

Concentration Estimates for Emden-Fowler Equations with Boundary Singularities and Critical Growth

We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $Ω$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial Ω$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $Ω$ is star-shaped around 0). However, in contrast to the non-satisfacto…

黎曼流形上Lane-Emden方程的最优Liouville定理

Optimal Liouville theorems for the Lane-Emden equation on Riemannian manifolds

本文在具有非负Ricci曲率的Riemannian流形上，通过严格证明亚临界Lane-Emden-Fowler方程正解的不存在性，扩展了Gidas-Spruck和Serrin-Zou的工作。

分数Hardy不等式的定量稳定性：无重排技术与Emden-Fowler分析

Quantitative stability for fractional Hardy inequalities: Rearrangement-free techniques and Emden-Fowler analysis

本文研究分数Hardy不等式的定量改进，通过局部Poincaré-Sobolev不等式和Lorentz嵌入，建立距离与不等式缺额之间的关系，改进了稳定性指数，并在局部情况下应用Emden-Fowler对应关系和伪微分算子，推导了非局部Hardy缺陷与局部的一致性。

拟线性Hardy--Emden--Fowler方程的径向解：精确衰减率、Delaunay解与不存在性

Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence

本文研究拟线性Hardy--Emden--Fowler方程径向正解在无穷远处的行为，通过斜率变量将方程化为平面系统，完整刻画了四种参数区间下解的精确衰减率、Delaunay周期解的存在性及无解条件。

无标准临界指数的混合局部与非局部拉普拉斯算子的Lane-Emden方程

Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation

本文研究了结合经典与分数阶拉普拉斯算子的混合Lane-Emden方程，利用Lyapunov-Schmidt型约化方法证明在略低于临界Sobolev指数时解的存在性，揭示了局部与非局部算子相互作用导致传统临界指数缺失，首次给出临界指数对偶性失效的椭圆算子实例。

Global solvability criteria for some classes of nonlinear second order ordinary differential equations

The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler equation and to the Van der Pol type equation.

Combined effects in nonlinear problems arising in the study of anisotropic continuous media

The paper deals with the study of a Lane-Emden-Fowler equation with Dirichlet boundary condition and variable potential functions. The analysis developed in this paper combines monotonicity methods with variational arguments. Remark (April 21, 2020): Our results were later studied also by D.-P. Covei, Quasilinear problems with the competition between convex and concave nonlinearities and variable potentials, Internat. J. Math. 24:1 (2013), art. 1350005), arXiv:1104.4626v1.

锥中非线性积分算子的基本解与临界 Lane-Emden 指数

Fundamental solutions and critical Lane-Emden exponents for nonlinear integral operators in cones

本文研究锥形区域中完全非线性积分算子的齐次基本解，并据此建立 Lane-Emden-Fowler 方程超解的 Liouville 型定理及广义 Hopf 引理。

Joseph-Lundgren阈值附近的非径向稳定解

Nonradial stable solutions near the Joseph--Lundgren threshold

该研究针对超临界Lane-Emden方程，在第一Joseph-Lundgren区间构造了指数接近上端点的非径向稳定整体解，否定了Chan和Wei的径向性猜想，是Emden-Fowler方程非径向稳定解的首个非平凡例子。

Supercritical problems in domains with thin toroidal holes

In this paper we study the Lane-Emden-Fowler equation $$(P)_ε \{Δu+|u|^{q-2}u=0 \ \hbox{in}\ \mathcal D_ε, u=0 \ \hbox{on}\ \partial\mathcal D_ε.$$ Here $\mathcal D_ε= \mathcal D \setminus \{x \in \mathcal D \ : \ \mathrm{dist}(x,Γ_\ell)\le ε\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $Γ_\ell$ is an $\ell-$dimensional closed manifold such that $Γ_\ell \subset \mathcal D$ with $1\le \ell \le N-3$ and $q={2(N-\ell)\over N-\ell-2}$. We prove that, under some symmetry assumptions, the number of sign changing solutions to $(P)_ε$ increases as $ε$ goes to zero.

Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation $$ -Δu+ u=u^{2^*-1}+λu^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2<q<2^*$ and $λ>0$ is a parameter. It is known that as $λ\to 0$, after a rescaling the ground state solutions of the equation converge to a particular solution of the critical Emden-Fowler equation $-Δu=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension $N=…

Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities

In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane-Emden-Fowler equations with singular nonlinearities and subquadratic convection term. We study the combined effects of various terms involved in these problems: sublinear or superlinear nonlinearities, singular nonlinear terms, convecti…

R^n中奇异Q-曲率方程的唯一性与对称破缺

Uniqueness and Symmetry Breaking for Singular $Q$-curvature Equations in $\R^n$

该研究针对R^n中奇异Q-曲率方程，通过变换建立解的对称性，证明径向解的唯一性与非退化性，构造非径向解分支，还证明γ足够小时所有解均为径向解。

Asymptotics for positive singular solutions to subcritical sixth order equations

We classify the local asymptotic behavior of positive singular solutions to a class of subcritical sixth order equations on the punctured ball. Initially, using a version of the integral moving spheres technique, we prove that solutions are asymptotically radially symmetric solutions with respect to the origin. We divide our approach into some cases concerning the growth of nonlinearity. In general, we use an Emden--Fowler change of variables to translate our problem to a cylinder. In the lower critical regime, this is not enough, thus, we need to introduce a new notion of change of variables.…

Nonlocal scalar field equations: qualitative properties, asymptotic profiles and local uniqueness of solutions

We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-Δ)^s u+εu &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \] where $s\in(0,1)$, $N>2s$, $q>p>2$ are fixed parameters and $ε>0$ is a vanishing parameter. For $ε>0$ small, we prove the existence of a ground state solution and show that any positive solution of above problem is a classical solution and radially symmetric and symmetric decreasing. We also obtain the decay rate of solution at infinity. Next, we study the asymptotic be…

Qualitative properties for solutions to conformally invariant fourth order critical systems

We study qualitative properties for nonnegative solutions to a conformally invariant coupled system of fourth order equations involving critical exponents. For solutions defined in the punctured space, there exist essentially two cases to analyze. If the origin is a removable singularity, we prove that non-singular solutions are rotationally invariant and weakly positive. More precisely, they are the product of a fourth order spherical solution by a unit vector with nonnegative coordinates. If the origin is a non-removable singularity, we show that the solutions are radially symmetric and stro…

Asymptotics for singular solutions to conformally invariant fourth order systems in the punctured ball

We study the asymptotic behavior for singular solutions to a critical fourth order system generalizing the constant $Q$-curvature equation. Our main result extends to the case of strongly coupled systems, the celebrated asymptotic classification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1989)] and [N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Invent. Math., (1999)]. On the technical level, we use an involved spectral analysis to study the Jacobi fields' growth properties in the kernel of the linearization of our system around a blow-up limit solution. Beside…

具有混合非线性项的分数阶 Schrödinger 方程：基态的渐近剖面、唯一性与非退化性

Fractional Schrödinger equations with mixed nonlinearities: asymptotic profiles, uniqueness and nondegeneracy of ground states

本文研究含消失参数的混合非线性分数阶 Schrödinger 方程，刻画次临界与临界情形下正基态解的渐近剖面，并据此证明小参数时解的唯一性与非退化性。

Classification for positive singular solutions to critical sixth order equations

We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity, solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients. On the technical level, we combine integral sliding methods and qualitative analysis of ODEs, based on a conservation of energy result, to perform a topological two-parameter shooting technique. We first use the integral representation …

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

We study the leading order behaviour of positive solutions of the equation -Δu +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\R^N, where $N\ge 3$, $q>p>2$ and when $\varepsilon>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $p^\ast=\frac{2N}{N-2}$. For $p<p^\ast$ the solution asymptotically coincides with the solution of the equation in which the last term is absent. For $p>p^\ast$ the so…

Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)Δu+ λu= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $λ\to 0$ and $λ\to +\infty$, where $N=3$ or $N= 4$, $2<q\le p\le 2^*$, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $a>0$, $b\ge 0$ are constants and $λ>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $λ\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-Δu+u=u^{q-1}$ a…

临界锥中奇异方程 $-Δu=u^{-γ}$ 的尖锐边界增长率估计

Sharp Boundary Growth Rate Estimate of the Singular Equation $-Δu=u^{-γ}$ in a Critical Cone

本文研究临界锥中奇异方程 $-Δu=u^{-γ}$ 的边界增长率，通过构造参考点序列将 PDE 转化为离散积分方程递推，得到 $γ>1$ 时的尖锐估计并解决了此前遗留的可解性条件充分必要性问题。