Jerome Busca, Maria J. Esteban, Alex Quaas
In this paper we extend existing results concerning generalized eigenvalues of Pucci's extremal operators. In the radial case, we also give a complete description of their spectrum, together with an equivalent of Rabinowitz's Global Bifurcation Theorem. This allows us to solve equations involving Pucci's operators.
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We explicitly evaluate the principal eigenvalue of the extremal Pucci's sup--operator for a class of special plane domains, and we prove that, for fixed area, the eigenvalue is minimal for the most symmetric set.
Eigenvalue Inequalities for Fully Nonlinear Elliptic Equations via the Alexandroff-Bakelman-Pucci Method
本文利用Alexandroff-Bakelman-Pucci方法建立Dirichlet或Robin特征值问题中椭圆微分算子的特征值不等式，推广至完全非线性椭圆方程，给出拉普拉斯算子L^p范数紧下界及解梯度的L^∞界与C^3估计。
We present the proof of several inequalities using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, we give a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric …
We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng's eigenvalue comparison theorem for the Laplace-Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng's bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng's upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principa…
Shape Optimization for the Principal Eigenvalue of the Pucci Operator in Three Dimensions
本文研究三维空间中Pucci极值算子主特征值的形状优化问题，通过构造双金字塔域族并证明在固定体积约束下，对称无剪切配置是最小化体积归一化主特征值的唯一解。
In this paper we introduce some fully nonlinear second order operators defined as weighted partial sums of the eigenvalues of the Hessian matrix, arising in geometrical contexts, with the aim to extend maximum principles and removable singularities results to cases of highly degenerate ellipticity.nn
We consider two classes of nonlinear eigenvalue problems with double-phase energy and lack of compactness. We establish existence and non-existence results and related properties of solutions. Our analysis combines variational methods with the generalized Pucci-Serrin maximum principle.
In this paper we study the asymptotic behavior of the principal eigenvalues associated to the Pucci operator in bounded domain $Ω$ with Neumann/Robin boundary condition i.e. $\partial_n u=αu$ when $α$ tends to infinity. This study requires Lipschitz estimates up to the boundary that are interesting in their own rights.
Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator
本文研究了由Pucci极值算子支配的弱耦合非线性椭圆方程组的多个正解的存在性，通过上下解方法、固定点论证和分支技术，证明了参数μ足够大时解的存在性和多重性。
In this paper we extend some existence's results concerning the generalized eigenvalues for fully nonlinear operators singular or degenerate. We consider the radial case and we prove the existence of an infinite number of eigenvalues, simple and isolated. This completes the results obtained by the author with Isabeau Birindelli for the first eigenvalues in the radial case, and the results obtained for the Pucci's operator by Busca Esteban and Quaas and for the $p$-Laplace operator by Del Pino and Manasevich.
We study local bifurcation from an eigenvalue with multiplicity greater than one for a class of semilinear elliptic equations. We evaluate the exact number of bifurcation branches of non trivial solutions and we compute the Morse index of the solutions in those branches.
In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear eigenvalue problem. We also show the existence of multiple solutions of the nonlinear problem using variational methods.
We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and positively homogeneous. Examples of such operators include the Hamilon-Jacobi-Bellman operator and the Pucci extremal operators. In the case that the two principal half-eigenvalues are not equal, we show that the measures which achieve the minimum in this formula provide a partial characterization of the solvability of the corresponding Dirichlet problem at res…
We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in $\mathbb{R}^n$ satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) $p$-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope metho…
In this paper we study for the incompressible Euler equations the global structure of the bifurcation diagram for the rotating doubly connected patches near the degenerate case. We show that the branches with the same symmetry merge forming a small loop provided that they are close enough. This confirms the numerical observations done in the recent work [10]
In 2009 Loc and Schmitt established a result on sufficient conditions for multiplicity of solutions of a class of nonlinear eignvalue problems for the p-Laplace operator under Dirichlet boundary conditions, extending an earlier result of 1981 by Peter Hess for the Laplacian. Results on necessary conditions for existence were also established. In the present paper the authors extend the main results by Loc and Schmitt to the $Φ$-Laplacian. To overcome the difficulties with this much more general operator it was necessary to employ regularity results by Lieberman, a strong maximum principle by P…
In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature…
In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_Ω\vert \nabla u\vert^2\,dx\right)Δu=λu+h(x,u,λ)\,\,\text{in}\,\, Ω,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\,\,Ω. \end{array} \right.\nonumber \end{equation} Under some natural hypotheses on $h$, we show that $\left(aλ_1,0\right)$ is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of $λ$, in which there exist positive solutions for the above problem wit…
Bifurcation of double eigenvalues for Aharonov-Bohm operators with a moving pole
该研究针对含原点平面区域内带Dirichlet边界的Aharonov-Bohm算子，分析半整数环量下极点沿原点邻域直线移动时双重特征值的分岔行为，证明正测度锥内直线移动会引发分岔，并给出对称区域尤其是圆盘情形的精确结果。
We construct the global bifurcation curves, solutions versus level of harvesting, for the steady states of a diffusive logistic equation on a bounded domain, under Dirichlet boundary conditions and other appropriate hypotheses, when $a$, the linear growth rate of the population, is below $λ_2+δ$. Here $λ_2$ is the second eigenvalue of the Dirichlet Laplacian on the domain and $δ>0$. Such curves have been obtained before, but only for $a$ in a right neighborhood of the first eigenvalue. Our analysis provides the exact number of solutions of the equation for $a\leqλ_2$ and new information on the…
A celebrated result in bifurcation theory is that global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem when the operators involved are compact. In this paper a simple example is constructed which satisfies the regularity hypotheses of the global bifurcation theorem, and the eigenvalue has algebraic multiplicity one, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differe…
Eigenvalue Bifurcation in Doubly Nonlinear Problems with an Application to Surface Plasmon Polaritons
本文研究解与特征值参数均非线性的非自伴特征值问题，利用Lyapunov-Schmidt约化证明从简单孤立特征值分岔的结果，并应用于证明PT对称构型中横电表面等离子体极化激元的存在性。
In this paper we study the following eigenvalue boundary value problem for Monge-Ampère equations: {equation} \{{array}{l} \det(D^2u)=λ^N f(-u)\,\, \text{in}\,\, Ω, u=0,\,\text{on}\,\, \partial Ω. {array}. {equation} We establish the unilateral global bifurcation results for the problem with $f(u)=u^N+g(u)$ and $Ω$ being the unit ball of $\mathbb{R}^N$. More precisely, under some natural hypotheses on the perturbation function $g:\mathbb{R}\rightarrow\mathbb{R}$, we show that $(λ_1,0)$ is a bifurcation point of the problem and there are two distinct unbounded continua of one-sign solutions, wh…
In this paper, we analyze an eigenvalue problem for a quasi-linear elliptic operators involving Dirichlet boundary condition in an open smooth bounded set of $\mathbb{R}^N$. We investigate a bifurcation results (from trivial solution and from infinity) of an eigenvalue problem involving the $(p,2)$-Laplace operator, from the Fu\v cik spectrum of the Laplacian.
Bifurcation from multiple eigenvalues of rotating traveling waves on a capillary liquid drop
本文研究毛细作用下近似球形液滴自由边界问题中旋转行波剖面的分岔，证明线性化问题的任意重数特征值均为分岔点，并利用哈密顿结构和变分方法克服二维退化困难。
We extend the Caffarelli-Świech-Winter $C^{1,α}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.
This work deals with the focusing Nonlinear Schrodinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.
Bifurcation from bubbles in nonconvex cones
研究非凸锥中临界半线性椭圆方程的气泡分支问题，通过特征值分析和分支定理证明非径向解的全局存在性。
We study the one-dimensional Kirchhoff type equation $$ -(b + a\Vert u'\Vert^{2}) u''(x) = λu(x)^p, x \in I:= (-1,1), \enskip u(x) > 0, \enskip x\in I, \enskip u(\pm 1) = 0, $$ where $\Vert u'\Vert = \left(\int_I u'(x)^2 dx\right)^{1/2}$, $a > 0, b > 0, p> 0$ are given constants and $λ> 0$ is a bifurcation parameter. We establish the exact solution $u_λ(x)$ and complete shape of the bifurcation curves $λ= λ(ξ)$, where $ξ:= \Vert u_λ\Vert_\infty$. We also study the nonlinear eigenvalue problem $$ -\Vert u'\Vert^{p-1} u''(x) = μu(x)^p, x \in I, \enskip u(x) > 0, x\in I, \enskip u(\pm 1) = 0, $$ …
In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to $L^{\infty}$, which implies $C^{1,α}$ smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray-Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski ge…
Bifurcation for indefinite weighted $p$-laplacian problems with slightly subcritical nonlinearity
本文研究带变号权重的超线性 $p$-laplacian 椭圆边值问题，证明两个主特征值处正解从平凡解分岔，并借助 Orlicz 空间处理紧性，将 Drabek 结果推广至轻微次临界非线性。
We analyze global bifurcations along the family of radially symmetric vortices in the Gross--Pitaevskii equation with a symmetric harmonic potential and a chemical potential $μ$ under the steady rotation with frequency $Ω$. The families are constructed in the small-amplitude limit when the chemical potential $μ$ is close to an eigenvalue of the Schrödinger operator for a quantum harmonic oscillator. We show that for $Ω$ near $0$, the Hessian operator at the radially symmetric vortex of charge $m_{0}\in\mathbb{N}$ has $m_{0}(m_{0}+1)/2$ pairs of negative eigenvalues. When the parameter $Ω$ is i…
A codimension-three bifurcation, characterized by a pair of purely imaginary eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of a pair of weakly coupled nonlinear oscillators with Z_2 + Z_2 symmetry. The methodology is based on Arnold's ideas of versal deformations of matrices for the linear analysis, and Poincaré normal forms for the nonlinear analysis of the system. The stratified subvariety of primary bifurcations of codimensions one and two is identified in the parameter space. The analysis reveals different types of solutions in the state space, including equil…
In this paper, we consider the bifurcation problem for fractional Laplace equation \begin{eqnarray*} \begin{array}{ll} (-Δ)^{s} u = λu + f(λ,\,x,\,u)& \mbox{in }Ω, u = 0 &\mbox{in }\mathbb{R}^n\backslash Ω, \end{array} \end{eqnarray*} where $Ω\subset \mathbb{R}^n,\,n> 2s (0<s<1)$ is an open bounded subset with smooth boundary, $(-Δ)^{s}$ stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue $λ_1$ of the eigenvalue problem \begin{eqnarray*} \begin{gathered} (-Δ)^{s} v = λv\,\,\,\mbox{in}\,\,Ω, v = 0 \,\,\,\,\mbox{in}\,\,\,\,\math…
This manuscript extends the analysis of a much studied singularly perturbed three-component reaction-diffusion system for front dynamics in the regime where the essential spectrum is close to the origin. We confirm a conjecture from a preceding paper by proving that the triple multiplicity of the zero eigenvalue gives a Jordan chain of length three. Moreover, we simplify the center manifold reduction and computation of the normal form coefficients by using the Evans function for the eigenvalues. Finally, we prove the unfolding of a Bogdanov-Takens bifurcation with symmetry in the model. This l…
Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves
该研究针对二维近圆形毛细液滴自由边界问题，推导了圆周及一维平坦环面上的Craig-Sulem公式，证明其哈密顿结构与运动常数，通过多重特征值分岔得到角动量参数化的唯一对称旋转波解轨道。
In this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled, extending a bifurcation theorem due to J. López-Gómez. To this aim, we use several results, such that Fredholm operator of index $0$ theory, eigenvalues of elliptic operators and the Krein-Rutman theorem. Lastly, we apply this result to some particular systems arising from population dynamics and determine a region of existence of coexistence states.
We study the bifurcation diagram of the Onsager free-energy func- tional for liquid crystals with orientation parameter on the sphere in three dimensions. In particular, we concentrate on a very general class of two- body interaction potentials including the Onsager kernel. The problem is reformulated as a non-linear eigenvalue problem for the kernel operator, and a general method to find the corresponding eigenvalues and eigenfunctions is presented. Our main tools for this analysis are spherical harmonics and a special algorithm for computing expansions of products of spherical harmon- ics in…
Bifurcation results and multiple solutions for the fractional $(p,q)$-Laplace operators
本文研究带 Dirichlet 边界条件的分数阶 $(p,q)$-Laplace 算子之和所对应的非线性非局部特征值问题，证明了从平凡解及无穷远处的分岔结果，并利用变分方法获得多解存在性。
In the paper the asymptotic bifurcation of solutions to a parameterized stationary semilinear Schrödinger equation involving a potential of the Kato-Rellich type is studied. It is shown that the bifurcation from infinity occurs if the parameter is an eigenvalue of the hamiltonian lying below the asymptotic bottom of the bounded part of the potential. Thus the bifurcating solution are related to bound states of the corresponding Schrödinger equation. The argument relies on the use of the (generalized) Conley index due to Rybakowski and resonance assumptions of the Landesman-Lazer or sign-condit…
By means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.
We study an abstract equation in a reflexive Banach space, depending on a real parameter $λ$. The equation is composed by homogeneous potential operators. By analyzing the Nehari sets, we prove a bifurcation result. In some particular cases we describe the full bifurcation diagram, and in general, we estimate the parameter $λ_b$ for which the problem does not have non-zero solution when $λ>λ_b$. We give many applications to partial differential equations: Kirchhoff type equations, Schrödinger equations coupled with the electromagnetic field, Chern-Simons-Schrödinger systems and a nonlinear eig…
The purpose of this paper is to study weak solutions of a nonlinear Neumann problem considered on a ball. Assuming that the potential is invariant, we consider an orbit of critical points, i.e. we do not assume that critical points are isolated. We apply techniques of the equivariant analysis to examine bifurcations from the orbits of trivial solutions. We formulate sufficient conditions for local and global bifurcations, in terms of the right-hand side of the system and eigenvalues of the Laplace operator. Moreover, we characterise orbits at which the global symmetry-breaking phenomenon occur…
Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls
本文研究含奇异势的穿孔球上完全非线性一致椭圆方程的主特征值问题，分析不同γ参数下解的存在性与性质，并给出Pucci算子下的显式特征值，推广了拉普拉斯算子的Hardy–Sobolev常数。
We consider a class of nonlinear integro-differential operators and prove existence of two principal (half) eigenvalues in bounded smooth domains with exterior Dirichlet condition. We then establish simplicity of the principal eigenfunctions in viscosity sense, maximum principles, continuity property of the principal eigenvalues with respect to domains etc. We also prove an anti-maximum principle and study existence result for some nonlinear problem via Rabinowitz bifurcation-type results.
On determination of the bifurcation type for a free boundary problem modeling tumor growth
本文针对二维肿瘤生长自由边界模型的对称破缺分岔类型判定难题，通过小参数高阶展开与分岔方向计算，证明其所有对称破缺分岔均为叉形分岔，揭示了该模型二维与三维情形的行为差异。
The Allen-Cahn equation with nonlinear truncated Laplacians: description of radial solutions
本文研究带非线性截断拉普拉斯算子的Allen-Cahn方程，通过监测ODE状态切换量，近乎完整描述径向解，发现无界解且解不振荡，与拉普拉斯及Pucci算子情形相反。
We numerically study solitary waves in the coupled nonlinear Schrödinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change i…
Bifurcations of unstable eigenvalues for Stokes waves derived from conserved energy
针对无限深流体无旋重力波的Euler方程，严格证明Stokes波线性化运动方程的零特征值分岔出现在能量对陡度的极值点，且新不稳定特征值仅沿陡度增大方向产生。
In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(μ_k^ν(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^ν)^+$ and $(\mathcal{C}_{k}^ν)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^ν$ from $(μ_k^ν(p), 0)$, where $μ_k^ν(p)$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $ν\in\{+,-\}$…
This paper shows that the nonlinear periodic eigenvalue problem $${cases} -Δu + V(x) u - f(x,u) = λu, u \in H^1(\IR^N), {cases}$$ has a nontrivial branch of solutions emanating from the upper bound of every spectral gap of $-Δ+ V$. No convexity condition is assumed. The following result of independent interest is also proven: the direct sum $Y \oplus Z$ in $H^1(\IR^N)$ associated to a decomposition of the spectrum of $-Δ+V$ remains "topologically direct" in the $L^p$'s (in the sense that the projections from $Y+Z$ onto $Y$ and $Z$ are $L^p$-continuous).
We study symmetry properties of viscosity solutions of fully nonlinear uniformly elliptic equations. We show that if $u$ is a viscosity solution of a rotationally invariant equation of the form $F(x,D^2u)+f(x,u)=0$, then the operator $\mathcal{L}_u=\mathcal{M}^++\frac{\partial f}{\partial u}(x,u)$, where $\mathcal{M}^+$ is the Pucci's sup--operator, plays the role of the linearized operator at $u$. In particular, we prove that if $u$ is a solution in a radial bounded domain, if $f$ is convex in $u$ and if the principal eigenvalue of $\mathcal{L}_u$ (associated with positive eigenfunctions) in …
We provide fundamental properties of the first eigenpair for fractional $p$-Laplacian eigenvalue problems under singular weights, which is related to Hardy type inequality, and also show that the second eigenvalue is well-defined. We obtain a-priori bounds and the continuity of solutions to problems with such singular weights with some additional assumptions. Moreover, applying the above results, we show a global bifurcation emenating from the first eigenvalue, the Fredholm alternative for non-resonant problems, and obtain the existence of infinitely many solutions for some nonlinear problems …
We consider the nonlinear eigenvalue problem $[D(u(t))u(t)']' + λg(u(t)) = 0$, $u(t) > 0$, $t \in I := (0,1)$, $u(0) = u(1) = 0$, which comes from the porous media type equation. Here, $D(u) = pu^{2n} + \sin u$ ($n \in \mathbb{N}$, $p > 0$: given constants), $g(u) = u$ or $g(u) = u + \sin u$. $λ> 0$ is a bifurcation parameter which is a continuous function of $α= \Vert u_λ\Vert_\infty$ of the solution $u_λ$ corresponding to $λ$, and is expressed as $λ= λ(α)$. Since our equation contains oscillatory term in diffusion term, it seems significant to study how this oscillatory term gives effect to …
Principal eigenvalues for Fully non linear singular or degenerate operators in punctured balls
本文研究穿孔球中带奇异势的完全非线性退化或奇异椭圆方程，证明主特征值及特征函数的存在性、唯一性与正则性，并给出 Pucci 算子下临界参数的显式值，推广了 Hardy–Sobolev 常数。
We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bi…
Harnack Theory and Rigidity for Singular and Degenerate Fully Nonlinear Elliptic Equations with Hamiltonians
本文研究带哈密顿项的奇异与退化完全非线性椭圆方程粘性解的相关性质，建立其整体正则性、Harnack不等式、Liouville刚性及主特征值的统一理论。
This paper is devoted to the investigation of the following nonlocal dispersal equation $$ u_{t}(t,x)=\frac{D}{σ^m}\left[\int_ΩJ_σ(x-y)u(t,y)dy-u(t,x)\right]+f(t,x,u(t,x)), \quad t>0,\quad x\in\overlineΩ, $$ where $Ω\subset\mathbb{R}^{N}$ is a bounded and connected domain with smooth boundary, $m\in[0,2)$, $D>0$ is the dispersal rate, $σ>0$ characterizes the dispersal range, $J_σ=\frac{1}{σ^{N}} J\left(\frac{\cdot}σ\right)$ is the scaled dispersal kernel, and $f$ is a time-periodic nonlinear function of generalized KPP type. We first study the principal spectral theory of the linear operator a…
Numerical Approximation and Bifurcation Results for an Elliptic Problem with Superlinear Subcritical Nonlinearity on the Boundary
针对边界上超线性次临界非线性椭圆问题，提出有限差分与延拓结合的方法，通过主特征值分析确定分支点并追踪完整分支曲线，验证了理论结果。
We study steady bifurcation for the coupled system body-liquid consisting of a sphere freely falling in a Navier-Stokes liquid under the action of gravity. In particular we show that, under the assumption that for the bifurcating solution the translational velocity of the sphere is parallel to the gravity, bifurcation takes place provided 1 is a simple eigenvalue of a suitable linear operator and the transversality property holds.
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