Using Harnack's inequality and a scaling argument we study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $ζ\in \partialΩ\cup\{\infty\}$ for the quasilinear elliptic equation $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{ in } Ω,$$ where $Ω$ is a domain in $\mathbb{R}^d$, $d\geq 2$, $1<p<d$, and $A=(a_{ij})\in L_{\rm loc}^{\infty}(Ω; \mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. It is assumed that the potential $V$ belongs to a certain Wolff class and has a generalized Fuchsian-type singularity at an isolated point $ζ\in \partial Ω\cup \{\infty\}$.
This paper examines the decay properties of positive solutions for a family of fully nonlinear systems of integral equations containing Wolf potentials and Hardy weights. This class of systems includes examples which are closely related to the Euler-Lagrange equations for several classical inequalities such as the Hardy-Sobolev and Hardy-Littlewood-Sobolev inequalities. In particular, a complete characterization of the fast decaying ground states in terms of their integrability is provided in that bounded and fast decaying solutions are shown to be equivalent to the integrable solutions. In ge…
We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth case $0<q< p-1$, where $Δ_{p}u = \text{div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian, and $σ$ is a nonnegative measurable function (or measure) on $\mathbb{R}^n$. As an application, we give a necessary and sufficient condition for the existence of a positive solution $u \in L^{r}(\mathbb{R}^{n})$ ($0<r<\infty$) to this problem, which was open even in the case $…
In this paper, we establish pointwise estimates for supersolutions of quasilinear elliptic equations with structural conditions involving a generalized Orlicz growth in terms of a Wolff type potential. As a consequence, under the extra assumption, we obtain that the supersolutions satisfy a Harnack inequality and local Hölder continuity.
We prove existence results concerning equations of the type $-Δ_pu=P(u)+μ$ for $p>1$ and $F_k[-u]=P(u)+μ$ with $1\leq k<\frac{N}{2}$ in a bounded domain $Ω$ or the whole $\mathbb{R}^N$, where $μ$ is a positive Radon measure and $P(u)\sim e^{au^β}$ with $a>0$ and $β\geq 1$. Sufficient conditions for existence are expressed in terms of the fractional maximal potential of $μ$. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of $μ$. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff po…
We study quasilinear elliptic equations of the type $-Δ_{p} u = σu^{q} + μ\; \; \text{in} \;\; \bf{R}^n$ in the case $0<q< p-1$, where $μ$ and $σ$ are nonnegative measurable functions, or locally finite measures, and $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian. Similar equations with more general local and nonlocal operators in place of $Δ_{p}$ are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions $u$: $$ u(x) \approx (\bf{W}_p σ(x))^{\frac{p-q}{p-q-1}} + \bf{K}_{p,q} σ(x) + \bf{W}_p μ(x), \quad x \in \bf{R…
This article examines the properties of positive solutions to fully nonlinear systems of integral equations involving Hardy and Wolff potentials. The first part of the paper establishes an optimal existence result and a Liouville type theorem for the integral systems. Then, the second part examines the decay rates of positive bound states at infinity. In particular, a complete characterization of the asymptotic properties of bounded and decaying solutions is given by showing that such solutions vanish at infinity with two principle rates: the slow decay rates and the fast decay rates. In fact,…
We study the equation --div(A(x, u)) = g(x, u, u) + $μ$ where $μ$ is a measure and either g(x, u, u) $\sim$ |u| q 1 u||u| q 2 or g(x, u, u) $\sim$ |u| s 1 u + ||u| s 2. We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.
We discuss recent advances in the theory of quasilinear equations of the type $ -Δ_{p} u = σu^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $σ$ is a nonnegative measurable function, or measure, for the $p$-Laplacian $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$, as well as more general quasilinear, fractional Laplacian, and Hessian operators. Within this context, we obtain some new results, in particular, necessary and sufficient conditions for the existence of solutions $u \in \text{BMO}(\mathbb{R}^n)$, $u \in L^r_{\rm loc}(\mathbb{R}^n)$, etc., and prove an enhanced…
We study finite energy solutions to quasilinear elliptic equations of the type $$ -Δ_pu=σ\, u^q \quad \text{in } \mathbb{R}^n,$$ where $Δ_p$ is the $p$-Laplacian, $p>1$, and $σ$ is a nonnegative function (or measure) on $\mathbb{R}^n$, in the case $0<q < p-1$ ( below the "natural growth" rate $q=p-1$ ). We give an explicit necessary and sufficient condition on $σ$ which ensures that there exists a solution $u$ in the homogeneous Sobolev space $L_0^{1,p}(\mathbb{R}^n)$, and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff poten…
We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-Δ_p u=σ\, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $σ\ge 0$ is an arbitrary locally integrable function, or measure, and $Δ_pu={\rm div}(\nabla u|\nabla u|^{p-2})$ is the $p$-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation $$-Δ_p v \, = {b} \, \frac {|\nabla v|^{p}}{v} + σ\quad \text{on} \, \, \, \R^n,$$ where ${b}>0$. These results are new even…
In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane--Emden type with measure data on a Carnot group $\mathbb G$ of arbitrary step. The quasilinear part involves operators of the $p$-Laplacian type $Δ_{\mathbb G,\,p}\,$, $1<p<\infty$. These results are based on new a priori estimates of solutions in terms of nonlinear potentials of Th. Wolff's type. As a consequence, we characterize completely removable singularities, and prove a Liouville type theorem for supersolutions of quasilinear equations with source terms which has be…
In this paper we study the Hausdorff dimension of a elliptic measure $μ_{f}$ in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of $σ-$finite $n-1$ dimensional Hausdorff measure for $p>n$ and the same result holds for $p=n$ with an assumption on the boundary. We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension $\leq n-1-δ$ for $p\geq n$ for some $δ$ which depends on various …
Quasilinear Equations with Exponential Reactions and Measure Data on Carnot Groups
School of Mathematical Science and LPMC, Nankai University（南开大学数学科学学院和LPMC）；Chern Institute of Mathematics and LPMC, Nankai University（南开大学陈省身数学研究所和LPMC）
We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $ζ\in\partialΩ\cup\{\infty\}$ of the quasilinear elliptic equations $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{in } Ω\setminus\{ζ\},$$ where $Ω$ is a domain in $\mathbb{R}^d$ ($d\geq 2$), and $A=(a_{ij})\in L_{\rm loc}^{\infty}(Ω;\mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. The potential $V$ lies in a certain local Morrey space (depending on $p$) and has a Fuchsian-type isolated singularity at $ζ$.
In this paper, we are concerned with the quasilinear PDE with weight $$ -div A(x,\nabla u)=|x|^a u^q(x), \quad u>0 \quad \textrm{in} \quad R^n, $$ where $n \geq 3$, $q>p-1$ with $p \in (1,2]$ and $a \in (-n,0]$. The positive weak solution $u$ of the quasilinear PDE is $\mathcal{A}$-superharmonic and satisfies $\inf_{R^n}u=0$. We can introduce an integral equation involving the wolff potential $$ u(x)=R(x) W_{β,p}(|y|^au^q(y))(x), \quad u>0 \quad \textrm{in} \quad R^n, $$ which the positive solution $u$ of the quasilinear PDE satisfies. Here $p \in (1,2]$, $q>p-1$, $β>0$ and $0 \leq -a<pβ<n$. W…
We develop an existence, regularity and potential theory for nonlinear integrodifferential equations involving measure data. The nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional $p$-Laplacean operator with measurable coefficients. We introduce a natural function class where we solve the Dirichlet problem, and prove basic and optimal nonlinear Wolff potential estimates for solutions. These are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by Boccardo & Gallouët \cite{BG1, BG2} and Kilpe…
The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: $ -Δ_p u = u^q + μ$ and $F_k[-u] = u^q + μ$, $u > 0$, on $R^n$, or on a bounded domain $Ω$. Here $Δ_p$ is the p-Laplacian, and $F_k[u]$ is the $k$-Hessian defined as the sum of $k\times k$ principal minors of the Hessian matrix $D^2 u$; $μ$ is a nonnegative measurable function (or measure) on $Ω$. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an op…
We study quasilinear elliptic equations of the type $$-Δ_pu=σ\, u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $Δ_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $σ\ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar probl…
