Hans Hartikainen
We study a Dynamic Programming Principle related to the $p$-Laplacian for $1 < p < \infty$. The main results are existence, uniqueness and continuity of solutions.
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For differential inequalities with the $\infty$-Laplacian in the principal part, we obtain conditions for the absence of solutions in unbounded domains. Examples are given to demonstrate the accuracy of these conditions.
The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among other things that the limit of the eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.
We prove the existence of a principal eigenvalue associated to the $\infty$-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.
In this paper we prove a Hölder regularity estimate for viscosity solutions of inhomogeneous equations governed by the infinite Laplace operator relative to a frame of vector fields.
In this note we give three counter-examples which show that the Maximum Principle generally fails for classical solutions of a system and a single equation related to the $\infty$-Laplacian. The first is the tangential part of the $\infty$-Laplace system and the second is the scalar $\infty$-Laplace equation perturbed by a linear gradient term. The interpretations of the Maximum Principle for the system are that of the Convex Hull Property and also of the Maximum Principle of the modulus of the solution.
We study a free boundary optimization problem in heat conduction, ruled by the infinity-Laplace operator, with lower temperature bound and a volume constraint. We obtain existence and regularity results and derive geometric properties for the solution and the free boundaries.
The anisotropic $\infty$-Laplacian eigenvalue problem with Neumann boundary conditions
本研究借助粘性解分析带诺伊曼边界的各向异性$p$-拉普拉斯当$p\rightarrow\infty$时的极限问题，证明特征值与特征函数的几何性质，验证Szegö-Weinberger型不等式的成立。
We investigate the limiting behavior of solutions to the inhomogeneous $p$-Laplacian equation $-Δ_p u = μ_p$ subject to Neumann boundary conditions. For right hand sides which are arbitrary signed measures we show that solutions converge to a Kantorovich potential associated with the geodesic Wasserstein-$1$ distance. In the regular case with continuous right hand sides we characterize the limit as viscosity solution to an infinity Laplacian / eikonal type equation.
An $L^\infty$-variational problem involving the Fractional Laplacian
本文研究了在给定Dirichlet数据条件下，分数Laplacian相关 supremal 函数的绝对极小解的存在性和唯一性，并证明了极小解满足一个分数PDE。
In this paper we study the limit as $p\to \infty$ in the evolution problem driven by the $p-$Laplacian with dynamical boundary conditions. We prove that the natural energy functional associated with this problem converges to a limit in the sense of Mosco convergence and as a consequence we obtain convergence of the solutions to the evolution problems. For the limit problem we show an interpretation in terms of optimal mass transportation and provide examples of explicit solutions for some particular data.
Higher regularity estimates for solutions to $\infty$-Laplacian-type models
本文研究了非齐次$\infty$-拉普拉斯方程在内部临界点的高阶正则性，获得了优于有界强迫项模型的平滑性估计，并与障碍及死核型问题的经典结果建立了联系。
In this manuscript, we will study the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional $\wp(\cdot)-$Laplacian operator involving constant/variable exponent. In the case of constant exponents, under some appropriate conditions, we will study the existence of solutions and asymptotic behavior of solutions by employing the subdifferential approach and we will study the problem when $\wp$ goes to $\infty$. Already, for case the weighted fractional $\wp(\cdot)$-Laplacian operator, we will also study the asymptotic behavior of the problem soluti…
We consider overdetermined boundary value problems for the $\infty$-Laplacian in a domain $Ω$ of $\R^n$ and discuss what kind of implications on the geometry of $Ω$ the existence of a solution may have. The classical $\infty$-Laplacian, the normalized or game-theoretic $\infty$-Laplacian and the limit of the $p$-Laplacian as $p\to \infty$ are considered and provide different answers.
$C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem
本文研究含分数阶拉普拉斯算子$(-Δ)^s$（$0<s<1$）的单相自由边界问题，将平坦边界的正则性从已有的$C^{1,α}$提升至$C^∞$，推广了$(-Δ)^{1/2}$情形下的相关结果。
We prove existence and uniqueness of viscosity solutions to the degenerate parabolic problem $u_t = Δ_\infty^h u$ where $Δ_\infty^h$ is the $h$-homogeneous operator associated with the infinity-Laplacian, $Δ_\infty^h u = |Du|^{h-3} < D^2uDu,Du>$. We also derive the asymptotic behavior of $u$ for the problem posed in the whole space and for the Dirichlet problem with zero boundary conditions.
In this work we analyze the behavior of the spectrum of the peridynamic fractional $p$-Laplacian, $(-Δ_p)_δ^s$, under the limit process $δ\to0^+$ or $δ\to+\infty$. We prove spectral convergence to the classical $p$-Laplacian under a suitable scaling as $δ\to0^+$ and to the fractional $p$-Laplacian as $δ\to+\infty$.
In this work we introduce volume constraint problems involving the nonlocal operator $(-Δ)_δ^{s}$, closely related to the fractional Laplacian $(-Δ)^{s}$, and depending upon a parameter $δ>0$ called horizon. We study the associated linear and spectral problems and the behavior of these volume constraint problems when $δ\to0^+$ and $δ\to+\infty$. Through these limit processes on $(-Δ)_δ^{s}$ we derive spectral convergence to the local Laplacian and to the fractional Laplacian as $δ\to 0^+$ and $δ\to +\infty$ respectively, as well as we prove the convergence of solutions of these problems to sol…
The $p$-Laplacian operator $Δ_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$ is not uniformly elliptic for any $p\in(1,2)\cup(2,\infty)$ and degenerates even more when $p\to \infty$ or $p\to 1$. In those two cases the Dirichlet and eigenvalue problems associated with the $p$-Laplacian lead to intriguing geometric questions, because their limits for $p\to\infty$ or $p\to 1$ can be characterized by the geometry of $Ω$. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general $p\in[1,\infty]$. We repor…
In this paper we study a non-homogeneous Neumann problem, where the $p(x)$-Laplacian is involved and $p=\infty$ in a subdomain. By considering a suitable sequence $p_k$ of bounded variable exponents such that $p_k \to p$ and replacing $p$ with $p_k$ in the original problem, we prove the existence of a solution $u_k$ for each of those intermediate ones. We show that the limit of the $u_k$ exists and after giving a variational characterization of it, in the part of the domain where $p$ is bounded, we show that it is a viscosity solution in the part where $p=\infty$. Finally, we formulate the pro…
Improved regularity estimates for Hardy-Hénon-type equations driven by the $\infty$-Laplacian
本文研究带奇异权重与强吸收的 $\infty$-Laplacian Hardy-Hénon 型方程，证明粘性解的显式改进正则性指数与非退化性，并推广零障碍、死核结果及给出 Liouville 型应用。
\[ \left\{ \begin{array} [c]{lll} -\left( Δ_{p}+Δ_{q(p)}\right) u=λ_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})δ_{x_{u}} & \mathrm{in} & Ω\\ u=0 & \mathrm{on} & \partialΩ, \end{array} \right. \] where $x_{u}$ is the (unique) maximum point of $\left\vert u\right\vert ,$ $δ_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & N<q(p)<p\\ (1,\infty) & \mathrm{if} & N<p<q(p) \end{array} \right. \] and $λ_{p}>0$ is such that \[ \min\left\{ \frac{\left\Vert \nabla u\right\Vert _{\infty}}…
Infinitely many solutions for two generalized poly-Laplacian systems on weighted graphs
本研究利用基于山路定理和拓扑度理论的变分方法，证明了加权有限图及带Dirichlet边值的加权局部有限图上的两类广义多拉普拉斯系统，均存在能量分别趋于正负无穷的两列无穷多解。
In this paper we consider the PDE system of vanishing normal projection of the Laplacian for $C^2$ maps $u : \mathbb{R}^n \supseteq Ω\longrightarrow \mathbb{R}^N$: \[ [\![\mathrm{D} u]\!]^\bot Δu = 0 \ \, \text{ in }Ω. \] This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the $p$-Laplace system for all $p\in [2,\infty]$. For $p=\infty$, the $\infty$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calc…
In this work, given $p\in (1,\infty)$, we prove the existence and simplicity of the first eigenvalue $λ_p$ and its corresponding eigenvector $(u_p,v_p)$, for the following local/nonlocal PDE system \begin{equation}\label{Eq0} \left\{ \begin{array}{rclcl} -Δ_p u + (-Δ)^r_p u & = & \frac{2α}{α+β}λ|u|^{α-2}|v|^βu & \mbox{in} & Ω\\ -Δ_p v + (-Δ)^s_p v& = & \frac{2β}{α+β}λ|u|^α|v|^{β-2}v & \mbox{in} & Ω u& =& 0&\text{ on } & \mathbb{R}^N \setminus Ω v& =& 0&\text{ on } & \mathbb{R}^N \setminus Ω, \end{array} \right. \end{equation} where $Ω$$\subset$ $\mathbb{R}^N$ is a bounded open domain, $0<r, s<…
On the second eigenvalue of the infinity Laplacian with Robin boundary conditions
研究Robin边界条件下p-拉普拉斯算子第二特征值在p趋于无穷时的极限，证明其收敛于无穷拉普拉斯算子的第二特征值。
In this paper we consider in a bounded domain $Ω\subset \mathbb{R}^N$ with smooth boundary an eigenvalue problem for the negative $(p,q)$-Laplacian with a Steklov type boundary condition, where $p\in (1,\infty)$, $q\in (2,\infty)$ and $p\neq q$. A full description of the set of eigenvalues of this problem is provided, thus essentially extending a recent result by Abreu and Madeira [1] related to the $(p,2)$-Laplacian.
We study the eigenvalue problem for a system of fractional $p-$Laplacians, that is, $$ \begin{cases} (-Δ_p)^r u = λ\dfracαp|u|^{α-2}u|v|^β &\text{in } Ω,\vspace{.1cm} (-Δ_p)^s u = λ\dfracβp|u|^α|v|^{β-2}v &\text{in } Ω, u=v=0 &\text{in }Ω^c=\R^N\setminusΩ. \end{cases} $$ We show that there is a first (smallest) eigenvalue that is simple and has associated eigen-pairs composed of positive and bounded functions. Moreover, there is a sequence of eigenvalues $λ_n$ such that $λ_n\to\infty$ as $n\to\infty$. In addition, we study the limit as $p\to \infty$ of the first eigenvalue, $λ_{1,p}$, and we o…
We provide a unified strategy to show that solutions of dynamic programming principles associated to the $p$-Laplacian converge to the solution of the corresponding Dirichlet problem. Our approach includes all previously known cases for continuous and discrete dynamic programming principles, provides new results, and gives a convergence proof free of probability arguments.
In this paper we study the partial differential equation \begin{equation} \begin{split} \partial_tu &= k(t)Δ_αu - h(t)φ(u), u(0) &= u_0. \end{split} \end{equation} Here $Δ_α$ is the fractional Laplacian, $k,h:[0,\infty)\to[0,\infty)$ are continuous functions and $φ:\mathbb{R}\to[0,\infty)$ is a convex differentiable function. If $u_0\in C_b(\mathbb{R}^d)\cap L^1(\mathbb{R}^d)$ we prove that the above equation has a classical global solution and is non-negative if $u_0\geq0$. Imposing some restrictions on the parameters we prove that the mass $M(t)=\int_{\mathbb{R}^d}u(t,x)dx$, $t>0$, of the sy…
We prove $L^\infty$ bounds and estimates of the modulus of continuity of solutions to the Poisson problem for the normalized infinity and $p$-Laplacian, namely \[ -Δ_p^N u=f\qquad\text{for $n<p\leq\infty$.} \] We are able to provide a stable family of results depending continuously on the parameter $p$. We also prove the failure of the classical Alexandrov-Bakelman-Pucci estimate for the normalized infinity Laplacian and propose alternate estimates.
In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\rightarrow+\infty$ in De Giorgi sense. In particular, we analyzed the $Γ$-convergence of non-homogeneous Dirichlet boundary problem for fractional $p$-Laplacian in this approximation process, and proved that as $p\rightarrow+\infty$ the minimizers of fractional $p$-Laplacian with Dirichlet boundary $Γ$-converges to a minimizer of Hölder $\infty$-Laplacian under the same Dirichlet boundary condition. On the other hand, we first investigate the asymptotic behaviour of non-homogeneous fractional $p$-fun…
In this paper we study, in an open bounded set $Ω\subset\mathbb R^N$ with Lipschitz boundary $\partialΩ$, the Dirichlet problem for a nonlinear singular elliptic equation involving the $1$--Laplacian and a total variation term, that is, the inhomogeneous case of the equation appearing in the level set formulation of the inverse mean curvature flow. Our aim is twofold. On the one hand, we consider data belonging to the Marcinkiewicz space $L^{N,\infty}(Ω)$, which leads to unbounded solutions. So, we have to begin introducing the suitable notion of unbounded solution to this problem. Moreover, e…
In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-Δ)^αu=λa(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ α\in(0,1) $, $ N\ge 2 $, $λ>0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations invo…
On the comparison principle for a nonlocal infinity Laplacian
本文通过比较原理证明了非局部无穷拉普拉斯方程在有界区域上粘性解的唯一性。
We consider the Dirichlet problem for positive solutions of the equation $-Δ_p (u) = f(u)$ in a convex, bounded, smooth domain $Ω\subset\R^N$, with $f$ locally Lipschitz continuous. \par We provide sufficient conditions guarantying $L^{\infty} $ a priori bounds for positive solutions of some elliptic equations involving the $p$-Laplacian and extend the class of known nonlinearities for which the solutions are $L^{\infty} $ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains.
We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional $\infty$-Laplacian $Δ_\infty^s$ for $s\in (\frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland, C., Caffarelli, L. and Figalli, A., \textsl{Nonlocal Tug-of-War and the inifnity fractional Laplacian}, Comm. Pure Appl. Math., \textbf{65}, pp. 337--380, (2012)]. Our expansions are parametrised by the radius of the removed singularity $ε$, and allow for the identification of $Δ_\infty^sϕ(x)$ as the $ε^{2s}$-order coefficient of the deviation of the $ε$-ave…
In this paper, we prove new Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$Δ^γ_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-γ} + f(x, u)\,=\,0\quad \text{in}\; \mathbb{R}^d,$$ where $γ\in [0, 2]$ and $Δ^γ_\infty$ is a $(3-γ)$-homogeneous operator associated with the infinity Laplacian. Under the assumptions $\liminf_{|x|\to\infty}\lim_{s\to0}f(x,s)/s^{3-γ}>0$ and $q$ is a continuous function vanishing at infinity, we construct a positive bounded solution to the equation and if $f(x,s)/s^{3-γ}$ decreasing in $s$, we further obtain the u…
We study the behavior as $p\rightarrow\infty$ of $u_{p},$ a positive least energy solution of the problem \[ \left\{\begin{array} [c]{lll} \left[ \left( -Δ_{p}\right) ^α+\left( -Δ_{q(p)}\right) ^β\right] u=μ_{p}\left\Vert u\right\Vert _{\infty}^{p-2} u(x_{u})δ_{x_{u}} & \mathrm{in} & Ω\\ u=0 & \mathrm{in} & \mathbb{R}^{N}\setminusΩ\\ \left\vert u(x_{u})\right\vert =\left\Vert u\right\Vert _{\infty}, & & \end{array} \right. \] where $Ω\subset\mathbb{R}^{N}$ is a bounded, smooth domain, $δ_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q…
In this article we find necessary and sufficient conditions for the strong maximum principle and compact support principle for non-negative solutions to the quasilinear elliptic inequalities $$Δ_\infty u + G(|Du|) - f(u)\,\leq 0\quad \text{in}\; \mathcal{O},$$ and $$Δ_\infty u + G(|Du|) - f(u)\,\geq 0\quad \text{in}\; \mathcal{O},$$ where $\mathcal{O}$ denotes the infinity Laplacian, $G$ is an appropriate continuous function and $f$ is a nondecreasing, continuous function with $f(0)=0$.
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical $p$-laplacian equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in elliptic regularity for the $p$-laplacian combined either with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a quasilinear boundary value problem $ -Δ_p u= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $p<N,$ is a bounded smooth domain strictly convex, and …
Let $Ω$ be a $C^\infty$-smooth bounded domain of $\mathbb{R}^n$, $n \geq 1$, and let the matrix ${\bf a} \in C^\infty (\overlineΩ;\R^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(Ω)$-realization $A$ of the operator $-\mydiv ( {\bf a} \nabla \cdot)$ with Dirichlet boundary conditions. We perturb $A$ by some real valued potential $V \in C_0^\infty (Ω)$ and note $A_V=A+V$. We compute the asymptotic expansion of $\mbox{tr}\left( e^{-t A_V}-e^{-t A}\right)$ as $t \downarrow 0$ for any matrix ${\bf a}$ whose coefficients are homogeneous of degree $0$. In the particular case where…
Global $L^\infty$ and decay estimate for fractional $p$-Laplacian equations in $D^{s,p}(\R^N)$
本文针对分数阶 $p$-Laplacian 方程提出新的全局 $L^\infty$ 估计，并利用非线性 Wolff 势证明衰减估计，进而得出关于局部极小值的 Brezis-Nirenberg 型结果。
This paper is concerned with the initial boundary value problem for one dimensional strongly damped wave equation involving $p$-Laplacian. For $p>2$, we establish the existence of weak local attractors for this problem in $W_{0}^{1,p}(0,1)\times L^{2}(0,1)$. Under restriction $2<p<4$, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in $W_{0}^{1,p}(0,1)\times L^{2}(0,1)$ and this attractor is a bounded subset of $W^{1,\infty }(0,1)\times W^{1,\infty }(0,1)$.
In this paper we study the Dirichlet eigenvalue problem $$ -Δ_p u-Δ_{J,p}u =λ|u|^{p-2}u \quad \text{ in } Ω,\quad u=0 \quad\text{ in } Ω^c=\mathbb{R}^N\setminusΩ. $$ Here $Δ_p u$ is the standard local $p-$Laplacian, $Δ_{J,p}u$ is a nonlocal, $p-$homogeneous operator of order zero and $Ω$ is a bounded domain in $\mathbb{R}^N$. We show that the first eigenvalue (that is isolated and simple) satisfies $(λ_1)^{1/p}\to Λ$ as $p\to\infty$ where $Λ$ can be characterized in terms of the geometry of $Ω$. We also find that the eigenfunctions converge, $u_\infty=\lim_{p\to\infty} u_p$, and find the limit…
On the solutions to $p$-Poisson equation with Robin boundary conditions when $p$ goes to $+\infty$
研究当 $p$ 趋于无穷大时带有 Robin 边界条件的 $p$-Laplacian 特征值及相关特征函数的极限行为，证明该极限是 $\infty$-Laplacian 特征值问题的粘性解。
Let $1<p \leq \infty$. We provide an asymptotic characterization of continuous viscosity solutions $u$ of the normalized $p$-Laplacian $Δ_{p\,\mathbb{G}}^N u=0$ in any Carnot group $\mathbb{G}$.
Denote by $Δ$ the Laplacian and by $Δ_\infty $ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $ΔvΔ_\infty v$: for every $v\in C^\infty$, $$\ | { |D^2vDv|^2} - {Δv Δ_\infty v } -\frac12[|D^2v|^2-(Δv)^2]|Dv|^2\ | \le \frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ Based on this, we prove the following results: 1. For any $p$-harmonic functions $u$, $p\in(1,2)\cup(2,\infty)$, we have $$|Du|^{\frac{p-γ}2}Du\in W^{1,2}_{\rm loc},$$ with $γ<\min\{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when $p\in(1,2)\cup(2,3+\frac2{n-2})$, we reprove the kno…
Let $Ω$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(Ω)$. Consider the functional $Q$ and its Gâteaux derivative $Q^\prime$ given by $$Q(u):= \frac{1}{p}\int_Ω. (|\nabla u|^p+V|u|^p) \dx, Q^\prime (u):= -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2} u.$$ In this paper we discuss a few aspects of relations between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^\prime (u)=0$.
Self-improving properties for the fractional $p$-Laplacian via nonlinear commutators
研究分数阶 $p$-Laplacian 和区域分数阶 $p$-Laplacian 弱解的自改进性质，通过扩展非线性交换子估计，证明了在非可积右端项和次二次情况下的局部正则性提升。
The asymptotic behavior of the first Robin eigenvalue with negative parameter as $p$ goes to $+\infty$
本文研究了带有负边界参数的第一 $p$-Laplacian Robin 特征值在 $p$ 趋向于正无穷时的渐近行为，并证明其特征函数的极限是 infinity Laplacian 特征值问题的粘性解。
On an evolutionary equation involving the nonlocal infinity Laplacian
研究非局部无穷拉普拉斯算子的演化方程，在柱体区域上利用 Perron 方法证明 Dirichlet 问题解的存在性和唯一性。
The purpose of this paper is to investigate the time behavior of the solution of a weighted $p$-Laplacian evolution equation, given by \begin{align} \label{eveq} \begin{cases} u_{t} = \text{div} \left(γ|\nabla u|^{p-2}\nabla u \right) & \text{on} (0,\infty)\times S, \\ γ|\nabla u|^{p-2}\nabla u\cdotη=0 & \text{on} (0,\infty)\times \partial S, \\ u(0,\cdot)=u_{0} & \text{on} S,\end{cases} \end{align} where $n \in \mathbb{N}\setminus \{1\}$, $p \in (1,\infty)\setminus \{2\}$, $S\subseteq \mathbb{R}^{n}$ is an open, bounded and connected set of class $C^{1}$, $η$ is the unit outer normal on $\par…
We study the behavior of solutions for the parametric equation $$-Δ_{p}^{a_1} u(z)-Δ_{q}^{a_2} u(z)=λ|u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } Ω,\, λ>0,$$ under Dirichlet condition, where $Ω\subseteq \mathbb{R}^N$ is a bounded domain with a $C^2$-boundary $\partial Ω$, $a_1,a_2 \in L^\infty(Ω)$ with $a_1(z),a_2(z)>0$ for a.a. $z \in Ω$, $p,q \in (1,\infty)$ and $Δ_{p}^{a_1},Δ_{q}^{a_2}$ are weighted versions of $p$-Laplacian and $q$-Laplacian. We prove existence and nonexistence of nontrivial solutions, when $f(z,x)$ asymptotically as $x \to \pm \infty$ can be resonant. In the studied cases…
In this paper we consider in a bounded domain $Ω\subset \mathbb{R}^N$ with smooth boundary an eigenvalue problem for the negative $(p,q)$-Laplacian with a Steklov-like boundary condition, where $p,\, q\in (1,\infty)$, $p\neq q$, including the open case $p\in (1,\infty)$, $q\in (1, 2)$, $p\neq q$. A full description of the set of eigenvalues of this problem is provided. Our results complement those previously obtained by Abreu and Madeira \cite{AM}, Barbu and Moroşanu \cite{BM}, Fărcăşeanu, Mihăilescu and Stancu-Dumitru \cite{FMS}, Mihăilescu \cite{MMih}, Mihăilescu and Moroşanu \cite{MM}.
This paper discusses the unique continuation principal of the solutions of the following perturbed fourth order elliptic differential operator $\mathcal{L}_{A,q}u=0$, where \[ \mathcal{L}_{A,q}(x,D)\ =\ \sum_{j=1}^nD^4_{x_j} + \sum_{j=1}^n A_jD_{x_j} + q, \qquad (A, q) \in W^{1,\infty}(Ω,\mathbb{C}^n) \times L^{\infty}(Ω,\mathbb{C}) \] whose principal term is not given by some integer power of the Laplacian operator. We derive some suitable Carleman estimates which is the main tool to prove the unique continuation principle. As a by-product, we also deduce some stability estimate and prove the…
We consider the parabolic, initial value problem $$ v_t =Δ_p(v)+λg(x,v)ϕ_p(v), \quad \text{in $Ω\times (0,\infty),$} $$ \[ v =0, \text{in $\partialΩ\times (0,\infty),$}\tag{IVP} v =v_0\ge0, \text{in $Ω\times \{0\},$} \] where $Ω$ is a bounded domain in ${\mathbb R}^N$, for some integer $N\ge1$, with smooth boundary $\partialΩ$, $ϕ_p(s):=|s|^{p-1} {\rm sgn}s$, $s\in{\mathbb R}$, $Δ_p$ denotes the $p$-Laplacian, with $p>\max\{2,N\}$, $v_0\in C^0(\overlineΩ)$, and $λ>0$. The function $g:\overline{Ω} \times [0,\infty)\to(0,\infty)$ is $C^0$ and, for each $x\in\overline{Ω}$, the function $g(x,\cdot…
We study the asymptotic behavior as $p\to\infty$ of the Gelfand problem \[ -Δ_{p} u=λ\,e^{u}\ \textrm{in}\ Ω\subset\mathbb{R}^n,\quad u=0 \ \textrm{on}\ \partialΩ. \] Under an appropriate rescaling on $u$ and $λ$, we prove uniform convergence of solutions of the Gelfand problem to solutions of \[ \min\left\{|\nabla{}u|-Λ\,e^{u}, -Δ_{\infty}u\right\}=0\ \textrm{in}\ Ω,\quad u=0\ \text{on}\ \partialΩ. \] We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of $Λ$.
This paper concerns the existence of global solutions for the following class of heat equation involving the 1-Laplacian operator of the Dirichlet problem $$ \left\{ \begin{array}{llc} u_{t}-Δ_1 u=f(u) & \text{in}\ & Ω\times (0, +\infty) , u =0 & \text{in} & \partialΩ\times (0, +\infty), u(x,0)=u_{0}(x)& \text{in} &Ω, \end{array}\right. \leqno{(P)} $$ where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $N \geq 1$ and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying some technical conditions, and $Δ_1 u=\text{div}\left(\frac{Du}{|Du|}\right)$ denotes the 1-Laplacian operat…
In a recent paper D. D. Hai showed that the equation $ -Δ_{p} u = λf(u) \mbox{in} Ω$, under Dirichlet boundary condition, where $Ω\subset {\bf R^N}$ is a bounded domain with smooth boundary $\partialΩ$, $Δ_{p}$ is the p-Laplacian, $f : (0,\infty) \rightarrow {\bf R} $ is a continuous function which may blow up to $\pm \infty$ at the origin, admits a solution if $λ> λ_0$ and has no solution if $0 < λ< λ_0$. In this paper we show that the solution set $\mathcal{S}$ of the equation above, which is not empty by Hai's results, actually admits a continuum of positive solutions.
We investigate the boundedness and large time behavior of solutions of the Cauchy-Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity: $$ u_t = (|u-x|^{p-2} u-x)_x+|u_x|^q \qquad \text{in}\quad (0, +\infty)\tiles(0, 1),\qquad q > p > 2.$$ We prove that: either $u_x$ blows up in finite time, or $u$ is global and converges in $W^{1, \infty}$norm to the unique steady state. This in particular eliminates the possibility of global solutions with unbounded gradient. For that purpose a Lyapunov functional is constructed by the approach of Zelenyak.
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