Umberto Guarnotta, Roberto Livrea, Salvatore Angelo Marano
A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided.
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In this paper we study the existence of positive smooth solutions for a class of singular (p(x),q(x))- Laplacian systems by using sub and supersolution methods.
Some recent existence, multiplicity, and uniqueness results for singular p-Laplacian systems either in bounded domains or in the whole space are presented, with a special attention to the case of convective reactions. A extensive bibliography is also provided.
In this work, we study regularity properties for nonvariational singular elliptic equations ruled by the infinity Laplacian. We obtain optimal $C^{1,α}$ regularity along the free boundary. We also show existence of solutions, nondegeneracy properties and fine geometric estimates for the free boundary.
We study viscosity solutions to degenerate and singular elliptic equations of $p$-Laplacian type on Riemannian manifolds. The Krylov-Safonov type Harnack inequality for the $p$-Laplacian operators with $1<p<\infty$ is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear $p$-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.
This paper deals with existence of positive solutions for a class of quasilinear elliptic systems involving the $Φ$-Laplacian operator and convex-concave singular terms. Our approach is based on the generalized Galerkin Method along with perturbartion techniques and comparison arguments in the setting of Orlicz-Sobolev spaces
We study a class of fractional $p$-Laplacian problems with weights which are possibly singular on the boundary of the domain. We provide existence and multiplicity results as well as characterizations of critical groups and related applications.
We study the regularity of the solutions to initial-boundary value problems for N-systems of the p-Laplacian type, in $n\geq 3$ space variables, with square-integrable external forces in the space-time cylinder. So, the ellipticity coefficient remains unbounded. The singular case $μ=0 $ is covered.
Strong solutions to singular discontinuous $p$-Laplacian problems
本文研究了Dirichlet型$p$-拉普拉斯问题正强解的存在性，其反应项在零点奇异且高度不连续，仅要求不连续点集勒贝格测度为零。
We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.
On singular $p$-Laplacian problems with discontinuous convection terms
研究了具有在零处奇异且高度不连续的非线性反应项及不连续对流项的p-拉普拉斯问题，证明了正强解的存在性，采用局部论证和合适的测度理论结果。
In this paper, we study singular solutions of linear problems with fractional Laplacian. First, we establish Bôcher type theorems on a punctured ball via distributional approach. Then, we develop a few interesting maximum principles on a punctured ball. Our distributional approach only requires the basic local L-1 integrability. We also introduce several simple and useful lemmas, which enable us to unify the treatments for both Laplacian and fractional Laplacian. These theorems, lemmas and the methods introduced here can be adapted and applied in other situations.
We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by $p$-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.
On a fractional $p$-Laplacian problem in the whole space and with singular reaction
研究全空间上带有奇异反应项的分数阶 $p$-Laplacian 问题，利用截断论证、变分方法和先验估计，证明了正弱解的存在性及其在无穷远处逐点衰减。
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 …
A note on the existence of a unique positive entropy solution to a fractional Laplacian with singular nonlinearities
本文证明了带非线性奇异项及非负有界 Radon 测度源项的分数阶 Laplacian 问题存在唯一的正熵解。
We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-Δ)^su,\,u-φ\bigr\}=0$ in $\mathbb R^n$, for general obstacles $φ$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.
We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.
We consider the IBVP in exterior domains for the p-Laplacian parabolic system. We prove regularity up to the boundary, extinction properties for p \in ( 2n/(n+2) , 2n/(n+1) ) and exponential decay for p= 2n/(n+1) .
Existence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by the Φ-Laplacian operator and the reaction term can be non-monotone. The main tools employed are a local minimum theorem and the Mountain Pass theorem, together with the truncation technique. Global C^{1,τ} regularity of solutions is also investigated, chiefly via a priori estimates and perturbation techniques.
We consider the p-Laplacian system of N equations in n space variables, 1< p\leq 2, under the homogeneous Navier slip boundary condition. Furthermore, the gradient of the velocity is replaced by the, more physical, symmetric gradient. We prove W^{2, q} regularity, up to the boundary, under suitable assumptions on the couple p,q. The singular case μ= 0 is covered.
We consider a nonlinear elliptic equation driven by the Dirichlet $p$-Laplacian with a singular term and a $(p-1)$-linear perturbation which is resonant at $+\infty$ with respect to the principal eigenvalue. Using variational tools, together with suitable truncation and comparison techniques, we show the existence of at least two positive smooth solutions.
In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.
In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.
Laplacian Vanishing Theorem for Quantized Singular Liouville Equation
本文针对带量化奇异源的奇异Liouville方程建立拉普拉斯消失定理，给出非简单爆破情形下系数函数拉普拉斯趋于零的结论，是该类方程的首个二阶估计，具有重要应用价值。
The authors of this paper study singular phenomena(vanishing and blowing-up in finite time) of solutions to the homogeneous $\hbox{Dirichlet}$ boundary value problem of nonlinear diffusion equations involving $p(x)$-\hbox{Laplacian} operator and a nonlinear source. The authors discuss how the value of the variable exponent $p(x)$ and initial energy(data) affect the properties of solutions. At the same time, we obtain the critical extinction and blow-up exponents of solutions.
Fractional p-Laplacian Kirchhoff-type problem involving a singular term via Nehari manifold
本文利用Nehari流形和Ekeland变分原理，证明了含变号非线性与奇异项的分数阶p-Laplacian Kirchhoff型问题在次临界和临界情形下至少存在两个正解。
We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow-up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b to ensure blow-up of the solution in finite time.
We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: \[ \partial_t β(u)-Δ_p u\ni 0\quad\text{ for }\tfrac{2N}{N+1}<p<2, \] where $β$ is a maximal monotone graph with a jump at zero and $Δ_p$ is the $p$-Laplacian. Moreover, a logarithmic type modulus of continuity is quantified, which has been conjectured to be optimal.
In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^α, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om. \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > sp, s \in (0,1), \la >0, 0 < q \leq 1 $ and $α\le p^*_s-1$. We use variational methods to show the existence and multiplicity of positive solutions of above problem with respect to parameter $\la$.
We study the existence of positive radially symmetric solution for the singular $p$-Laplacian Dirichlet problem, $-\bigtriangleup_p u =λ|u|^{p-2} u-γu^{-α}$ where $λ>0,γ>0$ and, $0<α<1$, are parameters and $Ω$, the domain of the equation, is a ball in $\mathbb{R}^N$. By using some variational methods we show that, if $λ$ is contained in some interval, then the problem has a radially symmetric positive solution on the ball. Moreover, we obtain a nonexistence result, whenever $λ\leq 0, γ<0$ and $Ω$ is a bounded domain, with smooth boundary.
In this paper local Lipschitz regularity of weak solutions to certain singular elliptic equations involving one-Laplacian is studied. Equations treated here also contains another well-behaving elliptic operator such as $p$-Laplacian with $1<p<\infty$. The problem is that one-Laplacian is too singular on degenerate points, what is often called facet, which makes it difficult to obtain even Lipschitz regularity of weak solutions. This difficulty is overcome by making suitable approximation schemes, and by avoiding analysis on facet for approximated solutions. The key estimate is a local a priori…
Existence, uniqueness, and decay results for singular $Φ$-Laplacian systems in $\mathbb{R}^N$
本文利用移位法、变分法及多种分析技术，研究了奇异 $\Phi$-Laplacian 系统解的存在性、先验估计、衰减性与唯一性，并给出了 Diaz-Saa 型结果。
Let $Ω$ be a bounded smooth domain in $\mathbb{R}^{4}$ such that for some integer $d\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem {Δ^{2}u-ρ^{4}k(x)e^{u}=0 & \hbox{in}Ω, u=Δu=0 & \hbox{on}\partialΩ, has a solution blowing-up, as $ρ\to0$, at $m$ points of $Ω$, for any given number $m$.
We study the behavior of positive solutions of p-Laplacian type elliptic equations of the form Q'(u) := -p-Laplacian(u) + V |u|^(p-2) u = 0 in Omega near an isolated singular point zeta, where 1 < p < inf, Omega is a domain in R^d with d > 1, and zeta = 0 or zeta = inf. We obtain removable singularity theorems for positive solutions near zeta. In particular, using a new three-spheres theorems for certain solutions of the above equation near zeta we prove that if V belongs to a certain Kato class near zeta and p>d (respectively, p<d), then any positive solution u of the equation Q'(u)=0 in a pu…
In this paper we prove the existence and uniqueness of positive classical solution of the fractional Laplacian with singular nonlinearity in a smooth bounded domain with zero Drichlet boundary conditions. By the method of sub-supersolution, we derive the existence of positive classical solution to the approximation problems. In order to obtain the regularity, we first establish the existence of weak solution for the fraction Laplacian. Thanks to \cite{XY}, the regularity follows from the boundedness of weak solution.
In this paper, we deal with existence, uniqueness and exact rate of boundary behavior of blow-up solutions for a class of logistic type quasilinear problem in a smooth bounded domain involving the $p$-Laplacian operator, where the nonlinearity can have a singular behavior. In the proof of the existence of solution, we have used the sub and super solution method in conjunction with variational techniques and comparison principles. Related to the rate on boundary and uniqueness, we combine a comparison principle proved in the present paper together with the our result of existence of solution.
Existence and regularity for a $p$-Laplacian problem in $\mathbb{R}^N$ with singular, convective, critical reaction
本文研究全空间 $\mathbb{R}^N$ 中带奇异、对流和临界反应项的 $p$-Laplacian 问题，结合变分法、截断、集中紧性、集值分析与不动点理论证明了解的存在性，并利用 De Giorgi 技术和正则性理论得到局部 $C^{1,α}$ 正则性与无穷远衰减。
Degenerate Schr{ö}dinger-Kirchhoff $(p, N)$-Laplacian problem With singular Trudinger-Moser nonlinearity in $\mathbb{R}^N$
本文研究ℝ^N中具有奇异指数非线性的退化(p, N)-Laplacian Schrödinger-Kirchhoff问题，利用山路定理、Ekeland变分原理、奇异Trudinger-Moser不等式及新的Brézis-Lieb型引理，证明了非平凡非负解的存在性。
This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $Δ_1 u$ and having a singular term of the type $\frac{f(x)}{u^γ}$. Here $f\in L^N(Ω)$ is nonnegative, $0<γ\le1$ and $Ω$ is a bounded domain with Lipschitz--continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions of $p$--Laplacian type problems. Moreover, when $f(x)>0$ a.e., the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit 1--dimensional exa…
A fractional Laplacian problem with mixed singular nonlinearities and nonregular data
本文研究了具有非正则数据和混合奇异非线性的分数阶拉普拉斯问题，证明了正解及熵解的存在性与唯一性，并通过放宽假设条件完善了存在性结果。
This paper is a continuation and an extension of our recent work [3] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In [3], the analytic behaviour of the Laplacian eigenfunctions is investigated at a point where two nodal or generalised singular lines intersect. It reveals a certain intriguing property that the vanishing order of the eigenfunction at the intersecting point is related to the rationality of the intersecting angle. In this paper, we consider the 3D counterpart of such a study by studying the analytic behaviours of th…
We consider the Brezis--Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both $\mathbb{R}^{n}$ and $\mathbb{S}^n$, $3 \le n \le 5$. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter ($δ$ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space i…
Fine boundary regularity for the singular fractional p-Laplacian
研究奇异分数阶p-Laplacian方程弱解的边界加权正则性，证明u/d_Ω^s有α-Hölder连续延拓，对应退化情形结果。
Regularity for degenerate/singular normalized $p$-Laplacian equations with Hamiltonian terms
针对变指数双相型退化/奇异和哈密顿项的规范化 $p$-Laplacian 方程，基于新的振荡型估计和局部分析，统一建立了尖锐的 $C^{1,α}$ 正则性估计，并在退化情形下得到了梯度 Hölder 正则性的改进结果。
We investigate the singular sets of solutions of conformally covariant elliptic operators of fractional order with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.
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…
Sharp regularity estimates for a singular inhomogeneous (m, p)-Laplacian equation
本文针对一类双重非线性发展偏微分方程，结合几何切向方法与内蕴尺度技术建立解在Hölder空间的精确正则性，统一处理三类扩散情形推广经典结果，并给出相关非线性模型的应用。
Positive normalized solutions for a singular regularized $p(x)$-Laplacian Dirichlet problem
该文研究带模约束的奇异正则化p(x)-Laplacian Dirichlet问题，通过构造非奇异逼近泛函的约束极小元，利用变分方法和Hardy不等式，证明了正归一化弱解的存在性。
On the singular problem involving $g$-Laplacian
本文研究了涉及分数阶 $g$-Laplacian 的奇异方程，在零 Dirichlet 条件下证明了其正弱解的存在性，并证明该解具有局部 Hölder 连续性，且该结果将混合分数阶 $(p,q)-$Laplacian 包含为特例。
We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-Δ)^γu=fu^p\quad\text{in }\mathbb R^n\setminusΣ$$ which are singular at $Σ$. Here $γ\in (0, \frac{n}{2})$. Among other things, we first prove that if $Σ$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d}<n-2γ$, and $ u\in L_γ(\mathbb R^n)\cap L^p_{loc}({\mathbb R^n\setminusΣ})$ is a non-negative solution for some $$p>\frac{n-\bf d}{n-{\bf d}-2γ},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solut…
We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$ \left\{\begin{array}{ll} \displaystyle -Δ_1 u =h(u)f(x)&\hbox{in }Ω\,,\\ u=0&\hbox{on }\partialΩ\,, \end{array}\right. $$ where $ Ω\subset \mathbb{R}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(Ω)$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions.
An equation involving the 1-Laplacian and a singular nonlinearity
研究具有奇异非线性项的1-Laplacian方程解的存在性，通过适当定义解的概念并利用逼近方法，证明了在边界条件下解的存在性。
We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation \begin{equation} -Δ_{p} u + \varepsilon u^{p-1} - u^{q-1} +u^{\mathit{l}-1} = 0 \qquad \text{in} \quad \mathbb{R}^{N}, \end{equation} where $1<p<N $, $p<q<l<+\infty$ and $\varepsilon> 0 $ is a small parameter. For $\varepsilon\rightarrow 0$, we give a characterisation of asymptotic regimes as a function of the parameters $q$, $l$ and $N$. In particular, we show that the behavior of the groundstates is sensitive to whether $q$ is less than, equal to, or greater than the critical Sobolev expon…
We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle -Δ_1 u = g(u)|D u|+h(u)f & \text{in}\;Ω,\\ u=0 & \text{on}\;\partialΩ, \end{cases} $$ where $Ω$ is an open bounded subset of $\mathbb{R}^N$, $f\geq 0$ belongs to $L^N(Ω)$, and $g$ and $h$ are continuous functions that may blow up at zero. As a noteworthy fact we show how a non-trivial interaction mechanism between the two nonlinearities $g$ and $h$ produces remarkable regularizing effects on the solutions. The sharpness of our main results is discusse…
In this paper, we study isolated singular positive solutions for the following Kirchhoff--type Laplacian problem: \begin{equation*} -\left(θ+\int_Ω |\nabla u| dx\right)Δu =u^p \quad{\rm in}\quad Ω\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial Ω, \end{equation*} where $p>1$, $θ\in \R$, $Ω$ is a bounded smooth domain containing the origin in $\R^N$ with $N\ge 2$. In the subcritical case: $1<p<N/(N-2)$ if $N\ge3$, $1<p<+\infty$ if $N=2$, we employ the Schauder fixed-point theorem to derive a sequence of positive isolated singular solutions for the above problem such that $M_θ(u)>0$. To es…
In this paper, we establish the results on the existence, nonexistence and multiplicity of positive solutions to singular boundary value problems involving $φ$-Laplacian. Our approach is based on the fixed point index theory. The interesting point is that a result for the existence of three positive solutions is given.
The purpose of this paper is to study nonlinear singular parabolic equations with $p(x)$- Laplacian. Precisely, we consider the following problem and discuss the existence of a non-negative weak solution. \begin{align*} \frac{\partial u}{\partial t}-Δ_{p(x)}u&=λu^{q(x)-1} + u^{-δ(x)}g+ f&&\text{in}~Q_T, u&= 0&&\text{on}~Σ_T, u(0,\cdot)&=u_0(\cdot)&&\text{in}~Ω\nonumber. \end{align*} Here $Q_T=Ω\times(0,T)$, $Σ_T=\partialΩ\times(0,T)$, $Ω$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz continuous boundary $\partialΩ$, $λ\in(0,\infty)$, $f\in L^1(Q_T)$, $g\in L^\infty(Ω)$, $u_0…
Existence and comparison results for a doubly singular 1-Laplacian problem with $L^1$ data
针对具L¹数据的双重奇异1-拉普拉斯问题，在极小假设下证明全局BV解存在性与比较原理，引入利用1-拉普拉斯算子刚性结构的新方法，可处理高度奇异项。
Eigenvalue for a problem involving the fractional (p,q)-Laplacian operator and nonlinearity with a singular and a supercritical Sobolev growth
本文通过变分法与上下解方法，研究分数阶 (p,q)-Laplacian 奇异椭圆特征值问题，涵盖超临界 Sobolev 增长，并建立解的 L∞ 范数估计。
Asymptotics of solutions to the porous medium equation near conical singularities
本文研究了具有锥奇点流形上多孔介质方程的渐近行为，证明其在锥点附近的解由锥截面Laplacian的谱决定，核心贡献在于精确描述了锥Laplacian的极大域。
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