We prove the uniqueness property for a class of entire solutions to the equation \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $\end{document} where $ \sigma $ is a nonnegative locally finite measure in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ {\rm div}\, \mathcal{A}(x, \nabla u) $ is the $ \mathcal{A} $-Laplace operator, under standard growth and monotonicity assumptions of order $ p $ ($ 1 < p < \infty $) on $ \mathcal{A}(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $); the model case $ \mathcal{A}(x, \xi) = \xi | \xi |^{p-2} $ corresponds to the $ p $-Laplace operator $ \Delta_p $ on $ \mathbb{R}^n $. Our main results establish uniqueness of solutions to a similar problem, \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma u^q +\mu, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $\end{document} in the sub-natural growth case $ 0 < q < p-1 $, where $ \mu, \sigma $ are nonnegative locally finite measures in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ \mathcal{A}(x, \xi) $ satisfies an additional homogeneity condition, which holds in particular for the $ p $-Laplace operator.

Abstract We study quasilinear elliptic equations of the type - Δ p ⁢ u = σ ⁢ u q + μ {-\Delta_{p}u=\sigma u^{q}+\mu} in ℝ n {\mathbb{R}^{n}} in the case 0 < q < p - 1 {0<q<p-1} , where μ and σ are nonnegative measurable functions, or locally finite measures, and Δ p ⁢ u = div ⁡ ( | ∇ ⁡ u | p - 2 ⁢ ∇ ⁡ u ) {\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian. Similar equations with more general local and nonlocal operators in place of Δ p {\Delta_{p}} are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions u: u ⁢ ( x ) ≈ ( 𝐖 p ⁢ σ ⁢ ( x ) ) p - q p - q - 1 + 𝐊 p , q ⁢ σ ⁢ ( x ) + 𝐖 p ⁢ μ ⁢ ( x ) , x ∈ ℝ n , u(x)\approx({\mathbf{W}}_{p}\sigma(x))^{\frac{p-q}{p-q-1}}+{\mathbf{K}}_{p,q}% \sigma(x)+{\mathbf{W}}_{p}\mu(x),\quad x\in\mathbb{R}^{n}, where 𝐖 p {{\mathbf{W}}_{p}} and 𝐊 p , q {{\mathbf{K}}_{p,q}} are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on p, q, and n. The contributions of μ and σ in these pointwise estimates are totally separated, which is a new phenomenon even when p = 2 {p=2} .

Bilateral pointwise estimates are provided for positive solutions u u to the sublinear integral equation u = G ( σ u q ) + f in Ω , \begin{equation*} u = \mathbf {G}(\sigma u^q) + f \quad \text {in } \ \Omega , \end{equation*} for 0 > q > 1 0 > q > 1 , where σ ≥ 0 \sigma \ge 0 is a measurable function or a Radon measure, f ≥ 0 f \ge 0 , and G \mathbf {G} is the integral operator associated with a positive kernel G G on Ω × Ω \Omega \times \Omega . The main results, which include the existence criteria and uniqueness of solutions, hold true for quasimetric, or quasimetrically modifiable kernels G G . As a consequence, bilateral estimates are obtained, along with existence and uniqueness, for positive solutions u u , possibly unbounded, to sublinear elliptic equations involving the fractional Laplacian, ( − Δ ) α 2 u = σ u q + μ in Ω , u = 0 in Ω c , \begin{equation*} (-\Delta )^{\frac {\alpha }{2}} u = \sigma u^q + \mu \quad \text {in}\quad \Omega , \quad u=0 \quad \text {in}\,\, \Omega ^c, \end{equation*} where 0 > q > 1 0>q>1 , and μ , σ ≥ 0 \mu , \sigma \ge 0 are measurable functions, or Radon measures, on a bounded uniform domain Ω ⊂ R n \Omega \subset \mathbb {R}^n for 0 > α ≤ 2 0 > \alpha \le 2 , or on the entire space R n \mathbb {R}^n , a ball or half-space, for 0 > α > n 0 > \alpha >n .

We give necessary and sufficient conditions for the existence of a BMO solution to the quasilinear equation $-\Delta_{p} u = \mu$ in $\mathbb{R}^n$, $u\ge 0$, where $\mu$ is a locally finite Radon measure, and $\Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian ($p>1$). We also characterize BMO solutions to equations $-\Delta_{p} u = \sigma u^{q} + \mu$ in $\mathbb{R}^n$, $u\ge 0$, with $q>0$, where both $\mu$ and $\sigma$ are locally finite Radon measures. Our main results hold for a class of more general quasilinear operators ${\rm div}(\mathcal{A}(x, \nabla \cdot))$ in place of $\Delta_{p}$.

We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type u = 𝐆(σ u^q) + f in Ω, where 0 < q < 1, σ≥ 0 is a Radon measure in Ω, f ≥ 0 is a measurable function, and 𝐆 is a linear integral operator with positive kernel G on Ω×Ω. For quasi-metric (or quasi-metrically modifiable) kernels G, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions u ∈ L^q_ loc (Ω, σ). Applications are considered to semilinear elliptic equations involving the (fractional) Laplacian, (-Δ)^α/2 u = σ u^q + μ in Ω, u=0 in Ω^c. Here 0<q<1, μ, σ≥ 0 are Radon measures, and Ω is a bounded uniform domain in ℝ^n, if 0 < α≤ 2, or the entire space ℝ^n, a ball or half-space, if 0 < α <n. Analogues of these results are presented for elliptic equations involving the p-Laplace operator on the entire space ℝ^n, -Δ_p u = σ u^q + μ in ℝ^n, lim inf_x→∞ u(x)=0, where 0<q<p-1, and μ, σ≥ 0 are Radon measures. More general quasilinear equations with 𝒜-Laplace operators div𝒜(x, ∇ u) in place of Δ_p are covered as well.

Let $$\Omega \subseteq \mathbb {R}^n$$ be an open set, where $$n \ge 2$$ . Suppose $$\omega $$ is a locally finite Borel measure on $$\Omega $$ . For $$\alpha \in (0,2)$$ , define the fractional Laplacian $$(-\triangle )^{\alpha /2}$$ via the Fourier transform on $$\mathbb {R}^n$$ , and let G be the corresponding Green’s operator of order $$\alpha $$ on $$\Omega $$ . Define $$T(u) = G(u \omega ).$$ If $$\Vert T \Vert _{L^2(\omega ) \rightarrow L^2 (\omega )} <1$$ , we obtain a representation for the unique weak solution u in the homogeneous Sobolev space $$L^{\alpha /2, 2}_0 (\Omega )$$ of $$\begin{aligned} (-\triangle )^{\alpha /2} u = u \omega + \nu \,\,\, \text{ on } \,\,\, \Omega , \,\,\, u=0 \,\,\, \text{ on } \,\,\, \Omega ^c, \end{aligned}$$ for $$\nu $$ in the dual Sobolev space $$L^{-\alpha /2, 2} (\Omega )$$ . If $$\Omega $$ is a bounded $$C^{1,1}$$ domain, this representation yields matching exponential upper and lower pointwise estimates for the solution when $$\nu = \chi _{\Omega }$$ . These estimates are used to study the existence of a solution $$u_1$$ (called the “gauge”) of the integral equation $$u_1=1+G(u_1 \omega )$$ corresponding to the problem $$\begin{aligned} (-\triangle )^{\alpha /2} u = u \omega \,\,\, \text{ on } \,\,\, \Omega , \,\,\, u \ge 0 \,\,\, \text{ on } \,\,\, \Omega , \,\,\, u=1 \,\,\, \text{ on } \,\,\, \Omega ^c . \end{aligned}$$ We show that if $$\Vert T \Vert <1$$ , then $$u_1$$ always exists if $$0<\alpha <1$$ . For $$1 \le \alpha <2$$ , a solution exists if the norm of T is sufficiently small. We also show that the condition $$\Vert T \Vert <1$$ does not imply the existence of a solution if $$1< \alpha <2$$ . The condition $$\Vert T \Vert \le 1$$ is necessary for the existence of $$u_1$$ for all $$0<\alpha \le 2$$ .

Abstract We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation - Δ p ⁢ u = σ ⁢ u q + μ   on ⁢ ℝ n -\Delta_{p}u=\sigma u^{q}+\mu\quad\text{on }\mathbb{R}^{n} in the sub-natural growth case 0 < q < p - 1 {0<q<p-1} , where Δ p {\Delta_{p}} ( 1 < p < ∞ {1<p<\infty} ) is the p-Laplacian, and σ, μ are positive Borel measures on ℝ n {\mathbb{R}^{n}} . Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case 0 < q < 1 {0<q<1} are treated for the fractional Laplace operator ( - Δ ) α {(-\Delta)^{\alpha}} in place of - Δ p {-\Delta_{p}} , on ℝ n {\mathbb{R}^{n}} for 0 < α < n 2 {0<\alpha<\frac{n}{2}} , and on an arbitrary domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} with positive Green’s function in the classical case α = 1 {\alpha=1} .

We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type \begin{equation*} u(x) - \int_\Omega G(x, y) \, g(u(y)) d \sigma (y) = h, \end{equation*} where $(\Omega, \sigma)$ is a locally compact measure space, $G(x, y)\colon \Omega\times \Omega \to [0, +\infty]$ is a kernel, $h \ge 0$ is a measurable function, and $g\colon [0, \infty)\to [0, \infty)$ is a monotone function. This problem is motivated by the semilinear fractional Laplace equation \begin{equation*} (-\Delta)^{\frac{\alpha}{2}} u - g(u) \sigma = \mu \quad \text{in} \, \, \Omega, \quad u=0 \, \, \, \text{in} \, \, \Omega^c, \end{equation*} with measure coefficients $\sigma$, $\mu$, where $g(u)=u^q$, $q \in \mathbb{R} \setminus\{0\}$, and $0<\alpha<n$, in domains $\Omega \subseteq\mathbb{R}^n$, or Riemannian manifolds, with positive Green's function $G$.

Let sigma, omega be measures on R-d, and let {lambda(Q)}(Q is an element of D) be a family of non-negative reals indexed by the collection D of dyadic cubes in R-d. We give necessary and sufficient conditions for the twoweight norm inequality parallel to T-lambda(f sigma)parallel to(Lq(omega)) <= C parallel to f parallel to(Lp)((sigma)) for every f is an element of L-p(sigma), for the positive dyadic operator T-lambda(f sigma):= Sigma(Q is an element of D) lambda(Q)(1/sigma(Q) integral(Q) fd sigma) 1(Q) in the difficult range 0 < q < 1 <= p < infinity of integrability exponents. This range of the exponents p, q appeared recently in applications to nonlinear PDE, which was one of the motivations for our study. Furthermore, we introduce a scale of discrete Wolff potential conditions that depends monotonically on an integrability parameter, and prove that such conditions are necessary (but not sufficient) for small parameters, and sufficient (but not necessary) for large parameters. Our characterization applies to Riesz potentials I-alpha(f sigma) = (-Delta)(-alpha/2) (f sigma), 0 < alpha < d, since it is known that they can be controlled by model dyadic operators. The weighted norm inequality for Riesz potentials in this range of p, q has been characterized previously only in the special case where sigma is a Lebesgue measure.

We study the existence problem for positive solutions $u \in L^{r}(\mathbb{R}^{n})$, $0

We give bilateral pointwise estimates for positive solutions of the equation { - u = ω u Ω, u ≥ 0, u = f ∂Ω , . in a bounded uniform domain Ω⊂ R^n, where ω is a locally finite Borel measure in Ω, and f≥ 0 is integrable with respect to harmonic measure d H^x on ∂Ω. We also give sufficient and matching necessary conditions for the existence of a positive solution in terms of the exponential integrability of M^* (m ω)(z)=∫_Ω M(x, z) m(x) d ω (x) on ∂Ω with respect to f d H^x_0, where M(x, ·) is Martin's function with pole at x_0∈Ω, m(x)=min (1, G(x, x_0)), and G is Green's function. These results give bilateral bounds for the harmonic measure associated with the Schrödinger operator - - ω on Ω, and in the case f=1, a criterion for the existence of the gauge function. Applications to elliptic equations of Riccati type with quadratic growth in the gradient are given.

In this note we prove Scurry's testing conditions for the boundedness of a sequence-valued averaging positive dyadic operator from a weighted Lp space to a sequence-valued weighted Lp space by using parallel stopping cubes.

We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth case $0

Let $\mathcal{L}$ be the general second order differential operator with complex-valued distributional coefficients $A=(a_{jk})_{j, k=1}^n$, $\vec{b}=(b_{j})_{j=1}^n$, and $c$ in an open set $\Omega \subseteq \mathbb{R}^n$ ($n \ge 1$), with principal part either in the divergence form, $\mathcal{L} u= {\rm div} \, (A \nabla u) + \vec{b} \cdot\nabla u + c \, u$, or non-divergence form, $ \mathcal L u= \sum_{j, \, k=1}^n \, a_{jk} \, \partial_j \partial_k u + \vec{b} \cdot\nabla u + c \, u $. We give a survey of the results by the authors which characterize the following two properties of $\mathcal{L}$: (1) $-\mathcal{L}$ is accretive, i.e., ${\rm Re} \, \langle -\mathcal L u, \, u\rangle \ge 0$; (2) $\mathcal L$ is form bounded, i.e., $\vert \langle \mathcal L u, u \rangle \vert \le C \, \Vert \nabla u \Vert_{L^2(\Omega)}^2$, for all complex-valued $u \in C^\infty_0(\Omega)$.

We obtain sharp pointwise estimates for positive solutions to the equation −Lu + Vuq = f, where L is an elliptic operator in divergence form, q ∈ ℝ\{0}, f ≥ 0 and V is a function that may change sign, in a domain Ω in ℝn, or in a weighted Riemannian manifold.

For the general second order linear differential operator $$\mathcal{L}_0=\sum_{j,k=1}^n{a_{jk}}\partial_j\partial_k+\sum_{j=1}^n{b_{j}}\partial_j+c$$ with complex-valued distributional coefficients aj,k, bj, and c in an open set Ω ⊆ ℝn (n ≥ 1), we present conditions which ensure that $$-\mathcal{L}_0$$ is accretive, i.e., Re $$\langle-\mathcal{L}_0\phi,\phi\rangle\geq0$$ for all φ ∈ C 0 ∞ (Ω).

We discuss recent advances in the theory of quasilinear equations of the type $ -\Delta_{p} u = \sigma u^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for the $p$-Laplacian $\Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$, 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 version of Wolff's inequality for intrinsic nonlinear potentials associated with such problems.

We study the behavior near the origin of C2 positive solutions u(x) and v (x) of the system $$\matrix{{0 \le - {\rm{\Delta}}u \le f(v)} \\ {0 \le - {\rm{\Delta}}v \le g(u)} \\} \quad {\rm{in}}\,{B_1}\left(0 \right)\,\backslash \left\{0\right\}\, \subset {\mathbb{R}^n},\,n \ge 2,$$ where f, g:(0, ∞) → (0, ∞) are continuous functions. We provide optimal conditions on f and g at ∞ such that solutions of this system satisfy pointwise bounds near the origin. In dimension n = 2 we show that this property holds if log+f or log+g grow at most linearly at infinity. In dimension n ≥ 3 and under the assumption f (t) = O(tλ), g(t) = O(tσ)as t → ∞ (λ, σ ≥ 0), we obtain a new critical curve that optimally describes the existence of such pointwise bounds. Our approach relies in part on sharp estimates of nonlinear potentials which appear naturally in this context.

Let L be the general second order differential operator with complex-valued distributional coefficients A = (ajk) n j,k=1, ~b = (bj) n j=1, and c in an open set Ω ⊆ R (n ≥ 1), with principal part either in the divergence form, Lu = div (A∇u)+~b ·∇u+ c u, or non-divergence form, Lu = ∑n j, k=1 ajk ∂j∂ku+ ~b · ∇u+ c u. We give a survey of the results by the authors which characterize the following two properties of L: (1) −L is accretive, i.e., Re 〈−Lu, u〉 ≥ 0; (2) L is form bounded, i.e., |〈Lu, u〉| ≤ C ‖∇u‖2L2(Ω), for all complex-valued u ∈ C∞ 0 (Ω).

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation ℒu = σ u^q + μ in Ω , in the sublinear case 0"0 . In this case u ∈ L^γ +q(Ω , σ )∩ L^γ(Ω , μ ) , where γ =1 corresponds to finite energy solutions. Here ℒu:= - div(𝒜∇ u) is a linear uniformly elliptic operator with bounded measurable coefficients, and σ , μ are nonnegative functions (or Radon measures), on an arbitrary domain Ω⊆ℝ^n which possesses a positive Green function associated with ℒ . When 0<γ≤ 1 , this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space Ẇ_0^1,p(Ω ) for 1"