By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta }_{p}u=a₁\left(x\right)F₁(x,u,v)$ in Ω, ⎪$-{\Delta }_{q}v=a₂\left(x\right)F₂(x,u,v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${ℝ}^N$ with smooth boundary or $\Omega ={ℝ}^N$. A nonexistence result is obtained for radially symmetric solutions.

We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $-div\left(a\right(x,u,\nabla u\left)\right)+g(x,u)=f-divF$ in Ω, where Ω is a bounded open domain of ${ℝ}^N$, N ≥ 2 and $a:\Omega \times ℝ\times {ℝ}^N\to {ℝ}^N$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to ${\prod }_{i=1}^N{L}^{p^{\text{'}}\left(·\right)}\left(\Omega \right)$.

ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces $H_{p,q;Y}(\Omega ,ℬ)$..........................................................73. Some properties of the spaces $H_{p,q;Y}(\Omega ,ℬ)$...................................................104. Some examples of the spaces $H_{p,q;Y}(\Omega ,ℬ)$....................................................155....

For $0<p-1<q$ and either $ϵ=1$ or $ϵ=-1$, we prove the existence of solutions of $-{\Delta }_{p}u=ϵu^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\partial C_S$, of the form $u\left(x\right)={\left|x\right|}^{-\beta }\omega (x/\left|x\right|)$. The problem reduces to a quasilinear elliptic equation on $S$ and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to $Lu=div\overrightarrow{f}$ in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the $L^q(\Omega ,d\mu )$ norm of |∇u| is dominated by the $L^p(\Omega ,dv)$ norms of $div\overrightarrow{f}$ and $|\overrightarrow{f}|$. If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation ${\sum }_{i=1}^m\frac{\partial }{\partial x_i}{a}_i(x,u,\nabla u)-c_0{\left|u\right|}^{p-2}u=f(x,u,\nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $\lambda \left(\right|u\left|\right){\sum }_{i=1}^m{a}_i(x,u,\eta ){\eta }_i\ge \nu \left(x\right){\left|\eta \right|}^p,$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$.

We consider a random, uniformly elliptic coefficient field $a$ on the lattice ${\mathbb{Z}}^d$. The distribution $\langle ·\rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\langle |{\nabla }_{x}G(t,x,y){{|}^2\rangle }^{1/2}$ and $\langle \left|{\nabla }_x{\nabla }_{y}G(t,x,y)\right|\rangle$. In particular, the elliptic Green’s function $G(x,y)$ satisfies optimal annealed bounds. In their recent work,...

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega$ is a smooth bounded domain of ${ℝ}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_p)$, as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta }_{p}u+{\left|u\right|}^{p-2}u=f_{1\lambda ₁}{\left(x\right)\left|u\right|}^{q-2}u+2\alpha /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }$, x ∈ Ω, ⎨ $-{\Delta }_{p}v+{\left|v\right|}^{p-2}v=f_{2\lambda ₂}{\left(x\right)\left|v\right|}^{q-2}v+2\beta /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {ℝ}^N$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i{\lambda }_i}\left(x\right)={\lambda }_i{f}_{i+}\left(x\right)+f_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where $f_{i\pm }\left(x\right)=max\pm f_i\left(x\right),0$, $g_{\mu }\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ denotes the p-Laplace operator. We use variational methods.

In this review article we present an overview on some a priori estimates in $L^p$, $p>1$, recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two $L^p$-bounds, $p>2$, for...

We study a general class of nonlinear elliptic problems associated with the differential inclusion $\beta \left(u\right)-div\left(a\right(x,Du)+F(u\left)\right)\ni f$ in Ω where $f\in L^{\infty }\left(\Omega \right)$. The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general $L^{\infty }$-data.

We study the existence of positive solutions to ⎧ ${div\left(\right|\nabla u|}^{p-2}\nabla u)+q\left(x\right)u^{-\gamma }=0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is ${ℝ}^N$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

We study nonlinear elliptic equations of the form $F\left(D^{2}u\right)=f\left(u\right)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $\xi \in {ℝ}^n$. We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.

We prove the existence of cylindrical solutions to the semilinear elliptic problem $-\Delta u+\frac{u}{{\left|y\right|}^2}=f\left(u\right)$, $u\in H^1\left({ℝ}^N\right)$, $u\ge 0$, where $(y,z)\in {ℝ}^k\times {ℝ}^{N-k}$, $N>k\ge 2$ and $f$ has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schr¨odinger and Klein–Gordon equations.

Our main purpose is to establish the existence of a positive solution of the system ⎧$-{∆}_{p\left(x\right)}u=F(x,u,v)$, x ∈ Ω, ⎨$-{∆}_{q\left(x\right)}v=H(x,u,v)$, x ∈ Ω, ⎩u = v = 0, x ∈ ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded domain with C² boundary, $F(x,u,v)={\lambda }^{p\left(x\right)}[g\left(x\right)a\left(u\right)+f\left(v\right)]$, $H(x,u,v)={\lambda }^{q(x})[g\left(x\right)b\left(v\right)+h\left(u\right)]$, λ > 0 is a parameter, p(x),q(x) are functions which satisfy some conditions, and $-{∆}_{p\left(x\right)}u=-{div\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)$ is called the p(x)-Laplacian. We give existence results and consider the asymptotic behavior of solutions near the boundary. We do not assume any symmetry conditions on the system.