Joon Hyuk Kang, Yun Myung Oh
In this paper, we give a sufficient condition for the existence, nonexistence and uniqueness of coexistence of positive solutions to a rather general type of elliptic competition system.
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We study existence, nonexistence, and uniqueness of positive radial solutions for a class of nonlinear systems driven by Pucci extremal operators under a Lane-Emden coupling configuration. Our results are based on the analysis of the associated quadratic dynamical system and energy methods. For both regular and exterior domain radial solutions we obtain new regions of existence and nonexistence. Besides, we show an exclusion principle for regular solutions, either in $\mathbb{R}^N$ or in a ball, by exploiting the uniqueness of trajectories produced by the flow. In particular, for the standard …
In this paper, we study the uniqueness of coexistence state for multiple species of animals in the same environment.
We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.
We study the following elliptic system with critical exponent: \begin{displaymath} \begin{cases}-Δu_j-\frac{λ_j}{|x|^2}u_j=u_j^{2^*-1}+\sum\limits_{k\neq j}β_{jk}α_{jk}u_j^{α_{jk}-1}u_k^{α_{kj}},\;\;x\in\R^N, u_j\in D^{1,2}(\R^N),\quad u_j>0 \;\; \hbox{in} \quad \R^N\setminus \{0\},\quad j=1,...,r.\end{cases}\end{displaymath} Here $N\geq 3, r\geq2, 2^*=\frac{2N}{N-2}, λ_j\in (0, \frac{(N-2)^2}{4})$ for all $ j=1,...,r $; $β_{jk}=β_{kj}$; \; $α_{jk}>1, α_{kj}>1,$ satisfying $α_{jk}+α_{kj}=2^* $ for all $k\neq j$. Note that the nonlinearities $u_j^{2^*-1}$ and the coupling terms all are critical…
We study the following quasilinear elliptic equation $$ -Δ_p u + (βΦ(x)-a(x)) u^{p-1} + b(x)g(u)=0 \quad \text{in } \mathbb{R}^N \quad \quad \quad (P_β) $$ where $p>1$, $Φ\in L^\infty_{loc}(\mathbb{R}^N)$, $a,b \in L^\infty(\mathbb{R}^N)$, $β,b,g \geq 0$. We provide a sharp criterion in terms of generalized principal eigenvalue for existence/nonexistence of positive solutions of ($P_β$) in suitable classes of functions. We derive the uniqueness result of ($P_β$) in those classes. Under additional conditions on $Φ$, we further show that : i) either for every $β\geq 0$ nonexistence phenomenon oc…
It is shown that third-order 1D nonlinear dispersion equations admit single point gradient catastrophe, described by blow-up-type similarity solutions. After blow-up, the solutions admit shock wave-type self-similar extensions. Snce such extensions are not unique, this implies the principle nonuniqueness of shock-type solutions and also nonexistence of any entropy-type description of proper unique solutions. A difficult free-boundary setting, with extra conditions specified on shocks, are necessary to restore uniqueness in such problems.
In this short paper I report on a paper published in Nonlinear Analysis: Real World Applications in 2004. There is a major mistake early in that paper which makes most of its claims false. The class of reaction-diffusion systems considered in the paper has been the object of a renewed investigation in the past few years, by myself and others, and recent discoveries provide explicit counterexamples .
In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth bounded domain. We prove the existence of weak solution in $W_{loc}^{s,p}(Ω)$ via approximation method. Establishing a new comparison principle of independent interest, we prove the uniqueness of weak solution for $0 \leq δ< 1+s- \frac{1}{p}$ and furthermore the nonexistence of weak solution for $δ\geq sp.$ Moreover, by virtue of barrier arguments we study t…
This paper explains the uniqueness of positive steady state of general Lotka-Volterra competition model of two species of animals in the same environment.
Consider classical solutions to the parabolic reaction diffusion equation $$ &u_t =Lu+f(x,u), (x,t)\in R^n\times(0,\infty); &u(x,0) =g(x)\ge0, x\in R^n; &u\ge0, $$ where $$ L=\sum_{i,j=1}^na_{i,j}(x)\frac{\partial^2}{\partial x_i \partial x_j}+\sum_{i=1}^nb_i(x)\frac\partial{\partial x_i} $$ is a non-degenerate elliptic operator, $g\in C(R^n)$ and the reaction term $f$ converges to $-\infty$ at a super-linear rate as $u\to\infty$. We give a sharp minimal growth condition on $f$, independent of $L$, in order that there exist a universal, a priori upper bound for all solutions to the above Cauch…
We discuss on existence, nonexistence and uniqueness of positive viscosity solutions for Lane-Emden systems involving the fractional Laplacian on bounded domains. As a byproduct, we obtain the critical hyperbole associated to the these systems.
Global existence and nonexistence analyses for a magnetic fractional pseudo-parabolic equation
本文研究磁分数阶Orlicz-Sobolev空间中伪抛物方程的初边值问题，利用嵌入定理、位势井理论和Galerkin方法证明了全局解的存在唯一性、衰减估计及有限时间爆破，并探讨了与平稳问题基态解的收敛关系。
We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on $M$ of the form $u_t=Δϕ(u)$, $ϕ$ being an arbitrary concave, increasing positive function, regular outside the origin and with $ϕ(0)=0$. Either property is also shown to be equivalent to nonexistence of nontrivial, nonnegative bounded solutions to the elliptic equation $ΔW=ϕ^{-1}(W)$ with $ϕ$ as above. As a consequence, ex…
This paper infers from a generalized Picone identity the uniqueness of the stable positive solution for a class of semilinear equations of superlinear indefinite type, as well as the uniqueness and global attractivity of the coexistence state in two generalized diffusive prototypes of the symbiotic and competing species models of Lotka-Volterra. The optimality of these uniqueness theorems reveals the tremendous strength of the Picone identity.
We study the existence or the nonexistence of classical solutions to a singular Gierer-Meinhardt system with Dirichlet boundary condition. The main feature of our model is that the activator and the inhibitor have different sources given by general nonlinearities. Additional regularity and uniqueness results are established for the one dimensional case.
Coexistence of heterogenous predator-prey systems with density-dependent dispersal
本文通过变量变换消除密度依赖扩散中的交叉扩散结构，结合指标理论与主特征值方法，对捕食者-猎物系统正解的存在性、非存在性与唯一性给出近乎完整分类，首次揭示密度依赖扩散通过提高捕食者存活机会促进物种共存。
In this paper, we investigate various properties (e.g., nonexistence, asymptotic behavior, uniqueness and integral representation formula) of positive solutions to nonlinear tri-harmonic equations in $\mathbb{R}^{n}$ ($n\geq2$) and bi-harmonic equations in $\mathbb{R}^{2}$ with negative exponents. Such kind of equations arise from conformal geometry.
We consider equation $-Δu+f(x,u)=0$ in smooth bounded domain $Ω\in\mathbb{R}^N$, $N\geqslant2$, with $f(x,r)>0$ in $Ω\times\mathbb{R}^1_+$ and $f(x,r)=0$ on $\partialΩ$. We find the condition on the order of degeneracy of $f(x,r)$ near $\partialΩ$, which is a criterion of the existence-nonexistence of a very singular solution with a strong point singularity on $\partialΩ$. Moreover, we prove that the mentioned condition is a sufficient condition for the uniqueness of a large solution and conjecture that this condition is also a necessary condition of the uniqueness.
Theoretical study of nonlinear wave equations with combined power-type nonlinearities with variable coefficients
本文研究变系数组合幂型非线性波动方程的初边值问题，完整刻画了非正与次临界能量下解的全局行为，确定了全局解不存在与非爆破解的阈值，并对超临界能量情形给出了两个新的全局解不存在的充分条件，同时证明了局部弱解的存在唯一性。
In this paper, we study strongly coupled elliptic systems in non-variational form with negative exponents involving fractional Laplace operators. We investigate the existence, nonexistence, and uniqueness of the positive classical solution. The results obtained here are a natural extension of the results obtained by Ghergu (2010), for the fractional case.
This paper studies the existence, nonexistence and uniqueness of positive solutions for a class of quasilinear equations. We also analyze the behavior of this solutions with respect to two parameters $κ$ and $λ$ that appears in the equation. The proof of our main results relies on bifurcation techniques, the sub and supersolution method and a construction of an appropriate large solutions.
In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schrödinger-Newton system. Precisely, we give the precise description of the chemical potential $μ$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local …
In this paper, we prove the N-barrier maximum principle, which extends the result in [5] from linear diffusion equations to nonlinear diffusion equations, for a wide class of degenerate elliptic systems of porous medium type. The N-barrier maximum principle provides a priori upper and lower bounds of the solutions to the above-mentioned degenerate nonlinear diffusion equations including the Shigesada-Kawasaki-Teramoto model as a special case. As an application of the N-barrier maximum principle to a coexistence problem in ecology, we show the nonexistence of waves in a three-species degenerate…
In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal.
In this paper we propose local and global existence results for the solution of systems characterized by the coupling of ODEs and PDEs. The coexistence of distinct mathematical formalisms represents the main feature of hybrid approaches, in which the dynamics of interacting agents are driven by second-order ODEs, while reaction-diffusion equations are used to model the time evolution of a signal influencing them. We first present an existence result of the solution, locally in time. In particular, we generalize the framework of recent works presented in the literature, concerning collective mo…
Nonlocal Diffusion Elliptic System Modelling The Behaviour Of a Bacteria And a Living Nutrient
本文研究模拟细菌与活性养料行为的非局部椭圆系统，通过分支方法和隐函数定理证明了共存态正解的存在性与唯一性。
We prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power-like zero order perturbations in bounded domains. The principal part of such equations is either $\mathcal{P}^-_{k}(D^2u)$ or $\mathcal{P}^+_{k}(D^2u)$, some sort of \lq\lq truncated Laplacians\rq\rq, given respectively by the smallest and the largest partial sum of $k$ eigenvalues of the Hessian matrix. New phenomena with respect to the semilinear case occur. Moreover, for $\mathcal{P}^-_{k}$, we explicitely find the critical exponent $p$ of the power nonlinear…
Fully nonlinear logistic equations with sanctuary
研究完全非线性 stationary 对数方程在有界域中的正解存在性与唯一性，分析参数 μ 对解的影响及解的渐进行为。
Some existence and uniqueness results for logistic Choquard equation
该文研究分数阶$p$-Laplacian驱动的双重非局部logistic Choquard方程，在一般非线性假设下分析弱解的存在性、唯一性等性质，应用于次扩散、超扩散问题并得到特定非线性项下能量变号解的存在性。
We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the Euclidean space, it is known from Herrero and Pierre [20] that the Cauchy problem associated with the FDE is well posed for initial data that are merely in $ L^1_{\mathrm{loc}} $. We establish here that such data still give rise to global solutions on general Riemannian manifolds. If, in addition, the radial Ricci curvature satisfies a suitable pointwise bound…
In this paper we study positive solutions to problem involving the fractional Laplacian $(E)$ $(-Δ)^α u(x)+|u|^{p-1}u(x)=0 in x\inΩ\setminus\mathcal{C}$, subject to the conditions $u(x)=0$ $x\inΩ^c$ and $\lim_{x\inΩ\setminus\mathcal{C}, x\to\mathcal{C}}u(x)=+\infty$, where $p>1$ and $Ω$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $\mathcal{C}\subset Ω$ is a compact $C^2$ manifold with $N-1$ multiples dimensions and without boundary, the operator $(-Δ)^α$ with $α\in(0,1)$ is the fractional Laplacian. We consider the existence of positive solutions for problem $(E)$. Moreover, we further …
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…
Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations
本文研究混合各向异性与非局部p-Laplace算子及双重非线性抛物方程，建立存在唯一性、比较定理、Hardy不等式及正则性估计，并利用Picone恒等式与能量估计等方法。
We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_λ$} \begin{cases} Δ^2u-μa(x)u=f(u)+λb(x)\quad\textrm{in $Ω$,}\\ u>0 \quad\textrm{in $Ω$,}\\ u=0=Δu \quad\textrm{on $\partialΩ$,} \end{cases} \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 5$, $a, b, f$ are nonnegaive functions satisfying certain hypothesis which we will specify later. $μ,λ$ are positive constants. Under some suitable conditions on functions $a, b, f$ and the constant $μ$, we show that there exists $λ^*>0$ such that when $0<λ<λ^*$, ($P_λ$) a…
Spectral-Edge Continuation, Dimensional Crossover, and Endpoint Stability for Partially Confined Nonlinear Schrödinger Equations
本文研究部分受限非线性薛定谔方程正解的全局几何与端点动力学，构造中心正解连通分支，揭示维数交叉，推导端点性质，为归一化驻波的存在性等提供依据。
Blow--up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources
本文研究了具有双曲动力边界条件的波方程的全局非存在性和爆炸性问题，考虑了内部和边界非线性阻尼与源项的影响。
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