Anna Lisa Amadori, Francesca Gladiali
In this paper we consider the problem $-Δu=|x|^α F(u)$ in $R^N$, with $α>0$ and $N\ge3$. Under some assumptions on $F$ we deduce the existence of nonradial solutions which bifurcate from the radial one when $α$ is an even integer.
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In this paper we are interested in positive classical solutions of \begin{equation} \label{eqx} \left\{\begin{array}{ll} -Δu = a(x) u^{p-1} & \mbox{ in } Ω, \\ u>0 & \mbox{ in } Ω, \\ u= 0 & \mbox{ on } \pOm, \end {array}\right. \end{equation} where $Ω$ is a bounded annular domain (not necessarily an annulus) in $\IR^N$ $(N \ge3)$ and $ a(x)$ is a nonnegative continuous function. We show the existence of a classical positive solution for a range of supercritical values of $p$ when the problem enjoys certain mild symmetry and monotonicity conditions. As a consequence of our results, we shall sh…
We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -Δu &=|x|^α|u|^{p-2}u&&\qquad \text{in ${\bf B}$,} \\ u&=0&&\qquad \text{on $\partial {\bf B}$,} \end{aligned} \right. \end{equation*} in the unit ball ${\bf B} \subset \mathbb{R}^N,N\geq 3$, $p>2$ in the limit $α\to +\infty$. More precisely, for a given positive integer $K$, we derive asymptotic $C^1$-expansions for the negative eigenvalues of the linearization of the unique radial solution $u_α$ of $(H)$ with precisely $K$ nodal domains and $u…
Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains
本文研究外部区域中的半线性椭圆方程，在权重满足特定对称与单调性条件下，证明了超临界指数范围内正解的存在性、非径向解的多重性及对称性破缺现象。
A supercritical elliptic equation in the annulus
本文在环域中研究超临界椭圆方程，利用变分与拓扑方法结合不变锥，构造了新的正轴对称解，并在常数系数情形下给出解非径向的充分条件。
Bifurcation and symmetry breaking for the Hénon equation
本文考虑球中的Hénon问题。我们证明存在（至少）一个分支的非径向解决方案，从径向的分支，这个分支是无界的。
Bifurcation diagrams of semilinear elliptic equations for supercritical nonlinearities in two dimensions
本文研究二维单位球中一般超临界非线性Gelfand问题，证明小参数下不稳定解不存在、径向奇异解存在及有限Morse指标解一致有界，从而得出分岔曲线具有无穷多个转折点，并阐明分岔结构仅由非线性超临界性决定。
We study radial solutions of the semilinear elliptic equation $Δu+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole-Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $Δu+u^p=0$ and $Δu+…
Monotonicity of the bifurcation curve for supercritical elliptic problems in the borderline dimension $N=10$
研究单位球上超临界半线性椭圆Dirichlet问题分支图的全局结构，在临界维度N=10处，通过构造满足Hardy型稳定性条件的奇异次解，证明了一类非线性（包括指数型与迭代指数函数）的分支曲线单调，从而得到经典解的唯一性，并刻画了分支曲线单调性与抛物问题全局无界解存在性的等价关系。
We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift-Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation. Using an approximation via modulation equations, we provide a tool to analyse how the noise influences the dynamics close to a change of stability.
Non-existence and uniqueness results for supercritical semilinear elliptic equations
证明了几个含参数的半线性椭圆型方程局部和非局部超临界分歧问题解的不存在性和唯一性。域是星形的，但不需要其他对称性假设。当分歧参数在一定范围内时，唯一性成立。我们的方法可以被看作是，在某些情况下，作为一个扩展的非平凡的解决方案的不存在的结果。它是基于Rellich-Pohozaev型估计。半线性椭圆型方程在许多应用中自然出现，例如天体物理学、流体力学或热力学。我们简化了K. Schmitt和R. Schaaf在所谓的局部乘法的情况下，将它们扩展到对分歧参数的非局部依赖的情况和添加剂的情况下，在局部和非局部设置。
In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -Δu=f(u)-μu, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $μ\in\mathbb{R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we deve…
We prove the existence of nonradial solutions for the Hénon equation in the ball with any given number of nodal zones, for arbitrary values of the exponent $α$. For sign-changing solutions, the case $α=0$ -- Lane-Emden equation -- is included. The obtained solutions form global continua which branch off from the curve of radial solutions $p\mapsto u_p$, and the number of branching points increases with both the number of nodal zones and the exponent $α$. The proof technique relies on the index of fixed points in cones and provides information on the symmetry properties of the bifurcating solut…
A parameter dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non-self-adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that e.g. can be interpreted as a thermostat device or in the context of an agent based price formation model…
Weakly nonlinear analysis of a two-species non-local advection-diffusion system
本文对包含非局部对流项的双物种对流扩散模型进行弱非线性分岔分析，推导出耦合零模态的 Ginzburg-Landau 振幅方程，揭示了不同参数下亚临界与超临界分岔的稳定性，并发现小振幅模式不稳定时会出现大振幅模式与滞后现象。
Semilinear elliptic problems on the half space with a supercritical nonlinearity
本文研究半空间上具Sobolev超临界非线性项和非齐次Dirichlet边界条件的标量场方程正解，在边界数据可积且非线性项满足Joseph--Lundgren次临界条件下，完整分类了正解的存在性，并通过分歧理论证明了多重正解的存在。
Sign-changing prescribed mass solutions for $L^2$-supercritical NLS on compact metric graphs
本文研究了在紧致度量图上临界质量非线性薛定谔方程的多变号解的存在性，首次在超临界质量范围内获得预设质量解的多重性结果，并证明了线性算子的特征值均为分支点。
We study the problem% \[ -Δv+λv=| v| ^{p-2}v\text{ in }Ω,\text{\qquad}v=0\text{ on $\partialΩ$},\text{ }% \] for $λ\in\mathbb{R}$ and supercritical exponents $p,$ in domains of the form% \[ Ω:=\{(y,z)\in\mathbb{R}^{N-m-1}\times\mathbb{R}^{m+1}:(y,| z| )\inΘ\}, \] where $m\geq1,$ $N-m\geq3,$ and $Θ$ is a bounded domain in $\mathbb{R}% ^{N-m}$ whose closure is contained in $\mathbb{R}^{N-m-1}\times(0,\infty)$. Under some symmetry assumptions on $Θ$, we show that this problem has infinitely many solutions for every $λ$ in an interval which contains $[0,\infty)$ and $p>2$ up to some number which i…
Normalized Solutions to Schrödinger Equations with General Nonlinearities in Bounded Domains via a Global Bifurcation Approach
该研究采用全局分岔方法，探究有界区域中具一般非线性项的薛定谔方程指定质量正解的存在性等性质，推广了已有自治齐次非线性情形的结果，并得到相关轨道稳定性结论。
We consider a semilinear elliptic problem in an annulus of R^N, with N>1. Recent results ensure that there exists a sequence p_k of exponents of the nonlinear term at which a nonradial bifurcation from the radial solution occurs. Exploiting the properties of O(N-1)-invariant spherical harmonics, we introduce two suitable cones K^1 and $K^2$ of O(N-1)-invariant functions that allow to separate the branches of bifurcating solutions from the others, getting the unboundedness of these branches.
On traveling wave solutions of water-wave equations in curved annular domains
本文研究弯曲环状区域中带常涡量二维欧拉方程的行波解，通过局部分岔分析确定临界参数并证明叉式分岔，再利用改进的 Leray-Schauder 度理论得到全局分岔结果，数值算例验证了亚临界与超临界两种分岔类型。
Bifurcation of finger-like structures in traveling waves of epithelial tissues spreading
本研究针对上皮单层铺展的连续活性极性流体模型，分析平前沿行波解的线性稳定性，证明以周期为分岔参数时手指状周期结构非平凡行波解的存在性，并明确收缩系数决定分岔为亚临界或超临界叉式分岔。
Ground states of critical and supercritical problems of Brezis-Nirenberg type
研究了超临界问题\[ -Δv=λv+\left\vert v\right\vert ^{p-2}v\text{ \ in }Ω,\qquad v=0\text{ on }\partialΩ, \]在\[ Ω=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert y\right\vert ,z\right) \inΘ\}, \]型区域中对称基态的存在性，其中$Θ$是有界光滑区域，使得$\overlineΘ \subset\left( 0,\infty\right) \times\mathbb{R}^{N-k-1},$ $1\leq k\leq N-3,$ $λ\in\mathbb{R},$和$p=\frac{2(N-k)}{N-k-2}$是$(k+1)$ -st临界指数.证明了$λ$在每个对称本征值左边的某个区间内存在对称基态，如果$k\geq2.$，则在$(-\infty,λ_{\ast})$与$λ_{\ast}>0$的某个区间内不存在对称基态。与此相关的问题是各向异性临界问题\[ -\text{div}(a(x)\nabla u)=λb(x)u+c(x)\left\vert u\right\vert ^{2^{\ast }-2}u\quad\text{in}\ Θ,\qquad u=0\quad\…
In this article, we study a one-dimensional nonlocal quasilinear problem of the form $u_t=a(\Vert u_x\Vert^2)u_{xx}+νf(u)$, with Dirichlet boundary conditions on the interval $[0,π]$, where $0<m\leq a(s)\leq M$ for all $s\in \mathbb{R}^+$ and $f$ satisfies suitable conditions. We give a complete characterization of the bifurcations and of the hyperbolicity of the corresponding equilibria. With respect to the bifurcations we extend the existing result when the function $a(\cdot)$ is non-decreasing to the case of general smooth nonlocal diffusion functions showing that bifurcations may be pitchf…
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