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

Applications of Mathematics

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

### A maximum principle for mean-curvature type elliptic inequalities

Journal of the European Mathematical Society

### A population biological model with a singular nonlinearity

Applications of Mathematics

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $\left\{\begin{array}{c}-{\mathrm{div}\left(|x|}^{-\alpha p}{|\nabla u|}^{p-2}{\nabla u\right)=|x|}^{-\left(\alpha +1\right)p+\beta }\left(a{u}^{p-1}-f\left(u\right)-\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 ${ℝ}^{N}$ with $0\in \Omega$, $1, $0\le \alpha <\left(N-p\right)/p$, $\gamma \in \left(0,1\right)$, and $a$, $\beta$, $c$ and $\lambda$ are positive parameters. Here $f:\left[0,\infty \right)\to ℝ$ 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...

### A weak comparison principle for some quasilinear elliptic operators: it compares functions belonging to different spaces

Applications of Mathematics

We shall prove a weak comparison principle for quasilinear elliptic operators $-\mathrm{div}\left(a\left(x,\nabla u\right)\right)$ that includes the negative $p$-Laplace operator, where $a:\Omega ×{ℝ}^{N}\to {ℝ}^{N}$ 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.

### An existence and approximation theorem for solutions of degenerate quasilinear elliptic equations

Commentationes Mathematicae Universitatis Carolinae

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.

### Averaging techniques and oscillation of quasilinear elliptic equations

Annales Polonici Mathematici

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

### Bernstein and De Giorgi type problems: new results via a geometric approach

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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$\phantom{\rule{0.166667em}{0ex}}\mathrm{div}\phantom{\rule{0.166667em}{0ex}}\left(a\left(|\nabla u\left(x\right)|\right)\nabla u\left(x\right)\right)+f\left(u\left(x\right)\right)=0.$Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in ${ℝ}^{2}$ and ${ℝ}^{3}$ and of the Bernstein problem on the flatness of minimal area graphs in ${ℝ}^{3}$. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach...

### Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains

Open Mathematics

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.

### Dirichlet-Neumann alternating algorithm for an exterior anisotropic quasilinear elliptic problem

Applications of Mathematics

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....

### Dynamic Programming Principle for tug-of-war games with noise

ESAIM: Control, Optimisation and Calculus of Variations

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...

### Entire solutions for a quasilinear problem in the presence of sublinear and super-linear terms.

Boundary Value Problems [electronic only]

### Entropy solutions for nonhomogeneous anisotropic ${\Delta }_{p⃗\left(·\right)}$ problems

Applicationes Mathematicae

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.

### Existence and multiplicity of positive solutions to a class of quasilinear elliptic equations in ${ℝ}^{N}$.

Boundary Value Problems [electronic only]

### Existence and multiplicity of solutions for divergence type elliptic equations.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Existence and symmetry of least energy solutions for a class of quasi-linear elliptic equations

Annales de l'I.H.P. Analyse non linéaire

### Existence and uniqueness of classical solutions to second-order quasilinear elliptic equations.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### Existence of a renormalized solution of nonlinear degenerate elliptic problems

Applicationes Mathematicae

We study a general class of nonlinear elliptic problems associated with the differential inclusion $\beta \left(u\right)-div\left(a\left(x,Du\right)+F\left(u\right)\right)\ni f$ 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.

### Existence of entropy solutions for degenerate quasilinear elliptic equations in ${L}^{1}$

Communications in Mathematics

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 \left(x\right)𝒜\left(x,u,\nabla u\right)\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 $G/\omega \in {\left[{L}^{{p}^{\text{'}}}\left(\Omega ,\omega \right)\right]}^{N}$.

### Existence of infinitely many weak solutions for some quasilinear $\vec {p}(x)$-elliptic Neumann problems

Mathematica Bohemica

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...

### Existence of multiple solutions for a $p\left(x\right)$-Laplace equation.

Electronic Journal of Differential Equations (EJDE) [electronic only]

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