### A continuity result for a quasilinear elliptic free boundary problem

We investigate a two dimensional quasilinear free boundary problem, and show that the free boundary is a union of graphs of continuous functions.

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We investigate a two dimensional quasilinear free boundary problem, and show that the free boundary is a union of graphs of continuous functions.

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$\left\{\begin{array}{c}-{\mathrm{div}\left(\right|x|}^{-\alpha p}{\left|\nabla u\right|}^{p-2}{\nabla u)=|x|}^{-(\alpha +1)p+\beta}\left(a{u}^{p-1}-f\left(u\right)-{\displaystyle \frac{c}{{u}^{\gamma}}}\right),\phantom{\rule{1.0em}{0ex}}x\in \Omega ,\hfill \\ u=0,\phantom{\rule{1.0em}{0ex}}x\in \partial \Omega ,\hfill \end{array}\right.$$ where $\Omega $ is a bounded smooth domain of ${\mathbb{R}}^{N}$ with $0\in \Omega $, $1<p<N$, $0\le \alpha <(N-p)/p$, $\gamma \in (0,1)$, and $a$, $\beta $, $c$ and $\lambda $ are positive parameters. Here $f:[0,\infty )\to \mathbb{R}$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions...

We shall prove a weak comparison principle for quasilinear elliptic operators $-\mathrm{div}\left(a\right(x,\nabla u\left)\right)$ that includes the negative $p$-Laplace operator, where $a:\Omega \times {\mathbb{R}}^{N}\to {\mathbb{R}}^{N}$ satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces.

The main result establishes that a weak solution of degenerate quasilinear elliptic equations can be approximated by a sequence of solutions for non-degenerate quasilinear elliptic equations.

By using averaging techniques, some oscillation criteria for quasilinear elliptic differential equations of second order ${\sum}_{i,j=1}^{N}{D}_{i}[{A}_{ij}{\left(x\right)\left|Dy\right|}^{p-2}{D}_{j}y]+p\left(x\right)f\left(y\right)=0$ are obtained. These results extend and generalize the criteria for linear differential equations due to Kamenev, Philos and Wong.

We use a Poincaré type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equations of the form$$\phantom{\rule{0.166667em}{0ex}}\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\left(a\left(\right|\nabla u\left(x\right)\left|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$$Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in ${\mathbb{R}}^{2}$ and ${\mathbb{R}}^{3}$ and of the Bernstein problem on the flatness of minimal area graphs in ${\mathbb{R}}^{3}$. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

We study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.

In this paper, by the Kirchhoff transformation, a Dirichlet-Neumann (D-N) alternating algorithm which is a non-overlapping domain decomposition method based on natural boundary reduction is discussed for solving exterior anisotropic quasilinear problems with circular artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation for the anisotropic quasilinear problems on circular artificial boundaries and construct the algorithm and analyze its convergence....

We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x ∈ Ω, Players I and II play an ε-step tug-of-war game with probability α, and with probability β (α + β = 1), a random point in the ball of radius ε centered at x is chosen. Once the game position reaches the boundary, Player II pays Player I the amount given by a fixed payoff function F. We give a detailed proof of the fact that...

We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.

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.

In this article, we prove the existence of entropy solutions for the Dirichlet problem $$\left(P\right)\left\{\begin{array}{cc}-\mathrm{div}\left[\omega \right(x\left)\mathcal{A}\right(x,u,\nabla u\left)\right]=f\left(x\right)-\mathrm{div}\left(G\right),\hfill & \text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=0,\hfill & \text{on}\phantom{\rule{4.0pt}{0ex}}\partial \Omega \hfill \end{array}\right.$$ where $\Omega $ is a bounded open set of ${}^{N}$, $N\ge 2$, $f\in {L}^{1}\left(\Omega \right)$ and $G/\omega \in {\left[{L}^{{p}^{\text{'}}}(\Omega ,\omega )\right]}^{N}$.

We consider the following quasilinear Neumann boundary-value problem of the type $$ \begin {cases} -\displaystyle \sum _{i=1}^{N}\frac {\partial }{\partial x_{i}}a_{i}\Big (x,\frac {\partial u}{\partial x_{i}}\Big ) + b(x)|u|^{p_{0}(x)-2}u = f(x,u)+ g(x,u) &\text {in} \ \Omega , \\ \quad \dfrac {\partial u}{\partial \gamma } = 0 &\text {on} \ \partial \Omega . \end {cases} $$ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev...