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 $p=2$.
Our version of Wolff's inequality for intrinsic nonlinear potentials relies on a new characterization of discrete Littlewood-Paley spaces $f^{p, q}(σ)$ defined in terms of characteristic functions of dyadic cubes in $\mathbb{R}^n$.
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 Fu…
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 reprove Wolff's $L^{\frac{5}2}-$ bound for the $\R^3-$Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right.
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.
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…
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 prove Harnack's inequality for bounded weak solutions to quasilinear second order elliptic equations with generalized Orlicz growth conditions. Our approach covers new cases of variable exponent and (p,q) growth conditions.
This paper is concerned with a class of quasilinear elliptic equations involving some potentials related to the Caffarelli-Korn-Nirenberg inequality. We prove the local boundedness and Hölder continuity of weak solutions by using the classical De Giorgi techniques. Our result extends the results of Serrin \cite{Serrin64} and Corolado and Peral \cite{CP04}.
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 extend some classical results dealing with boundary Harnack inequatilities to a class of quasilinear elliptic equations and derive some new estimates for solutions of such equations with an isolated singularity on the boundary of a domain.
We consider weak solutions to $$-Δ_pu+a(x,u)|\nabla u|^q=f(x,u),$$ with $p>1$, $q\geq\max\,\{p-1,1\}$. We exploit the Moser iteration technique to prove a Harnack comparison inequality for $C^1$ weak solutions. As a consequence we deduce a strong comparison principle.
We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,α}$ local estimates for solutions of a degenerate equation in non divergence form.
We study finite energy solutions to quasilinear elliptic equations of the type $$ -Δ_pu=σ\, u^q \quad \text{in } \mathbb{R}^n,$$ where $Δ_p$ is the $p$-Laplacian, $p>1$, and $σ$ is a nonnegative function (or measure) on $\mathbb{R}^n$, in the case $0<q < p-1$ ( below the "natural growth" rate $q=p-1$ ). We give an explicit necessary and sufficient condition on $σ$ which ensures that there exists a solution $u$ in the homogeneous Sobolev space $L_0^{1,p}(\mathbb{R}^n)$, and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff poten…
A weighted norm inequality of Muckenhoupt-Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains. This essentially leads to a resolution of an existence problem for quasilinear Riccati type equations with a gradient source term of arbitrary power law growth.
We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-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 work, we have proved a version of the Hardy-Littlewood-Sobolev inequality for variable exponents. After we use the variational method to establish the existence of solution for a class of Choquard equations involving the $p(x)$-Laplacian operator.
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 …
We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \begin{eqnarray} \text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for }x\inΩ\nonumber \end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local Hölder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness …
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）
Uniqueness of the strong positive solution for a general quasilinear elliptic problem with variable exponents and homogeneous Neumann boundary conditions using a generalization of the $p(x)$-Díaz-Saa inequality
We prove an existence result for a quasilinear elliptic equation satisfying natural growth conditions. As a consequence, we deduce an existence result for a quasilinear elliptic equation containing a singular drift. A key tool, in the proof, is the study of an auxiliary variational inequality playing the role of "natural constraint"
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 introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.
We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.
A weighted norm inequality involving $A_1$ weights is obtained at the natural exponent for gradients of solutions to quasilinear elliptic equations in Reifenberg flat domains. Certain gradient estimates in Lorentz-Morrey spaces below the natural exponent are also obtained as a consequence of our analysis.
We investigate an initial-boundary value problem for a quasilinear nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary condition of Silver & Müller type in a smooth, bounded, strictly star-shaped domain of $\mathbb{R}^{3}$. Imposing usual smallness assumptions in addition to standard regularity and compatibility conditions, a nonlinear stabilizability inequality is obtained by showing nonlinear dissipativity and observability-like estimates enhanced by an intricate regularity analysis. With the stabilizability inequality at hand, the classic nonlinear barrier method is …
In this article we are interested in addressing the question of existence and uniqueness of positive solution to a quasilinear elliptic equation involving p-laplacian in Heisenberg Group. The idea is to prove the uniqueness by using Diaz-Saa Inequality in Heisenberg Group which we obtain via a generalized version of Picone's Identity.
Using the framework first presented by Ruf and Sani, we give a proof of an Adams type inequality which can be applied to a nonlinear functional. Under two kinds of assumptions on the nonlinearity, we estimate the min-max level of the functional. As an application, a multiplicity result for the related singular quasilinear elliptic equation is proved.
We consider a combination of local and nonlocal $p$-Laplace equations and discuss several regularity properties of weak solutions. More precisely, we establish local boundedness of weak subsolutions, local Hölder continuity of weak solutions, Harnack inequality for weak solutions and weak Harnack inequality for weak supersolutions. We also discuss lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. Our approach is purely analytic and it is based on the De Giorgi-Nash-Moser theory, the expansion of positivity and estimates involving a tail term. The…
In this paper, we prove a sharp local well-posedness result for spherically symmetric solutions to quasilinear wave equations with rough initial data, when the spatial dimension is three or higher. Our approach is based on Morawetz type local energy estimates with fractional regularity for linear wave equations with variable $C^1$ coefficients, which rely on multiplier method, weighted Littlewood-Paley theory, duality and interpolation. Together with weighted linear and nonlinear estimates (including weighted trace estimates, Hardy's inequality, fractional chain rule and fractional Leibniz rul…
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…
Following the previous work [1], we investigate the impact of damping on the oscillation of smooth solutions to some kind of quasilinear wave equations with Robin and Dirichlet boundary condition. By using generalized Riccati transformation and technical inequality method, we give some sufficient conditions to guarantee the oscillation of all smooth solutions. From the results, we conclude that positive damping can ``hold back" oscillation. At last, some examples are presented to confirm our main results.
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation \[ u_t-\mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right)=-|u|^{β-1}u+α|u|^{q-2}u, \] where $p>1,β>0$, $q\geq1$ and $α>0$. By using Gagliardo-Nirenberg type inequality, energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.
We consider a quasilinear equation given in the half-space, i.e. a so called boundary reaction problem. Our concerns are a geometric Poincaré inequality and, as a byproduct of this inequality, a result on the symmetry of low-dimensional bounded stable solutions, under some suitable assumptions on the nonlinearities. More precisely, we analyze the following boundary problem $$ \left\{\begin{matrix} -{\rm div} (a(x,|\nabla u|)\nabla u)+g(x,u)=0 \qquad {on $\R^n\times(0,+\infty)$} -a(x,|\nabla u|)u_x = f(u) \qquad{\mbox{on $\R^n\times\{0\}$}}\end{matrix} \right.$$ under some natural assumptions o…
Let $f :\R\to\R$ be a continuous function. We prove that under some additional assumptions on $f$ and $A:\R\to\R_{+}$, weak $\Cuno$ solutions of the differential inequality $-\diver(A(\abs{\nabla u})\nabla u)\ge f(u)$ on $\RN$ are nonnegative. Some extensions of the result in the framework of subelliptic operators on Carnot Groups are considered.
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove $L^p$($p \ge 2$) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for w…
This note concerns an extension of the good-$λ$ inequality for fractional integrals, due to B. Muckenhoupt and R. Wheeden. The classical result is refined in two aspects. Firstly, general nonlinear potentials are considered; and secondly, the constant in the inequality is proven to decay exponentially. As a consequence, the exponential integrability of the gradient of solutions to certain quasilinear elliptic equations is deduced. This in turn is a consequence of certain Morrey space embeddings which extend classical results for the Riesz potential. In addition, the good-$λ$ inequality proved …
In a bounded smooth domain of Rn, we shall establish Kato's inequalities involving measures for p-Laplace operator up to the boundary under suitable assumptions. Keywords: Kato's inequality, $p$-Laplace operator
We study the enclosure method for the p-Calderón problem, which is a nonlinear generalization of the inverse conductivity problem due to Calderón that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.
This article focuses on almost global existence for quasilinear wave equations with small initial data in 4-dimensional exterior domains. The nonlinearity is allowed to depend on the solution at the quadratic level as well as its first and second derivatives. For this problem in the boundaryless setting, Hörmander proved that the lifespan is bounded below by $\exp(c/ε)$ where $ε>0$ denotes the size of the Cauchy data. Later Du, the second author, Sogge, and Zhou showed that this inequality also holds for star-shaped obstacles. Following up on the authors' work in the 3-dimensional case, we wea…
Assume that $p > 1$ and $p - 1 \le α\le p$ are real numbers and $Ω$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^α \ge 0, $$ where $D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$ is the gradient operator and $A : Ω\times {\mathbb R}^n \to {\mathbb R}^n$ and $b : Ω\to [0, \infty)$ are some functions with $$ C_1 |ξ|^p \le ξ A (x, ξ), \quad |A (x, ξ)| \le C_2 |ξ|^{p-1}, \quad C_1, C_2 = const > 0, $$ for almost all $x \in Ω$ and for all $ξ\in {\mathbb R}^n$. For solutions of this inequality we obtain…
We introduce a class of weak solutions to the quasilinear equation $-Δ_p u = σ|u|^{p-2}u$ in an open set $Ω\subset\mathbf{R}^n$. Here $p>1$, and $Δ_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored to general distributional coefficients $σ$ satisfying a certain weighted Sobolev-Poincare inequality. We also study weak solutions of the closely related equation $-Δ_p v = (p-1)|\nabla v|^p + σ$, under the same conditions on $σ$. Our results for this latter equation will allow us to characterize the class of distributions $σ$ which satisfy the Sobolev-Poincare inequality, there…
We prove the existence of solutions for an evolution quasi-variational inequality with a first order quasilinear operator and a variable convex set, which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable {\em a priori} estimates. We obtain also the existence of stationary solutions, by studying the asymptotic behaviour in time. In the variational case, correspond…
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…
In this article we study quasilinear multipower systems of two equations of two types, in a domain $Ω$ of R^{N} : with absorption terms, or mixed terms. Despite of the lack of comparison principle, we prove a priori estimates of Keller-Osserman type. Concerning the mixed system, we show that one of the solutions always satisfies Harnack inequality. In the case $Ω$=B(0,1)\{0}, we also study the behaviour near 0 of the solutions of more general weighted systems, giving a priori estimates and removability results. Finally we prove the sharpness of the results.
We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_α\ast u^p)u^q$ in $Ω$, where $Ω\subset \mathbb{R}^N, N\geq 1,$ is an open set. Here $I_α$ stands for the Riesz potential of order $α\in (0, N)$, $p>0$ and $q\in \mathbb{R}$. For a large class of operators $L_\mathcal{A}$ (which includes the $m$-Laplace and the $m$-mean curvature operator) we obtain optimal ranges of exponents $p,q$ and $α$ for which positive solutions exist. Our methods are then extended to quasilinear elli…
