Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent

In this work, we give an existence result of entropy solutions for nonlinear anisotropic elliptic equation of the type $$- \mbox{div} \big( a(x,u,\nabla u)\big)+ g(x,u,\nabla u) + |u|^{p_{0}(x)-2}u = f-\mbox{div} \phi(u),\quad \mbox{ in } \Omega,$$ where $-\mbox{div}\big(a(x,u,\nabla u)\big)$ is a Leray-Lions operator, $\phi \in C^{0}(I\!\!R,I\!\!R^{N})$. The function $g(x,u,\nabla u)$ is a nonlinear lower order term with natural growth with respect to $|\nabla u|$, satisfying the sign condition and the datum $f$ belongs to $L^1(\Omega)$.     

Local Continuity and Harnack’s Inequality for Double-Phase Parabolic Equations

Potential Analysis - We consider parabolic equations of the form $$ u_{t}-\text{div} \left( |\nabla u|^{p-2}\nabla u+ a(x,t)|\nabla u|^{q-2}\nabla u\right)= 0, a(x,t)\geq 0. $$ In the range $\frac...     

Existence of solutions for a degenerate auasilinear elliptic system in bounded domain

Using variational methods, we study the existence of weak solutions forthe degenerate quasilinear elliptic system$$\left\{\begin{array}{ll}- \mathrm{div}\Big(h_1(x)|\nabla u|^{p-2}\nabla u\Big) = F_{u}(x,u,v) &\text{ in } \Omega,\\-\mathrm{div}\Big(h_2(x)|\nabla v|^{q-2}\nabla v\Big) = F_{v}(x,u,v) &\text{ in } \Omega,\\u=v=0 & \textrm{ on } \partial\Omega,\end{array}\right.$$where $\Omega\subset \mathbb R^N$ is a smooth bounded domain, $\nabla F= (F_u,F_v)$ stands for the gradient of $C^1$-function $F:\Omega\times\mathbb R^2 \to \mathbb R$, the weights $h_i$, $i=1,2$ are allowed to vanish somewhere,the primitive $F(x,u,v)$ is intimately related to the first eigenvalue of acorresponding quasilinear system.     

Nonlinear superharmonic functions and supersolutions

Due to the lack of representation formulas for superharmonic functions associated with p-harmonic equations ${-\nabla \cdot(|\nabla u|^{p-2}\nabla u) = \mu}$ and their generalizations ${-\nabla \cdot A(x,\nabla u) = \mu}$ ,where ${A(x,\nabla u) \cdot \nabla u \approx | \nabla u |^{p}}$ , the interplay between nonlinear superharmonic functions and supersolutions is more important than in the linear case. Using the recent result of Kilpeläinen et. al., we establish sufficient and necessary conditions in terms of the Riesz measure μ that a p-superharmonic function is an ordinary weak supersolution. As an example we consider p-superharmonic solutions of the Poisson-type equation ${-\nabla \cdot A(x,\nabla u) = f(x)}$ .     

A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals

The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type

Existence of solutions of the Neumann problem for a class of equations involving the p-Laplacian via a variational principle of Ricceri

In this paper we deal with the existence of weak solutions for the following Neumann problem¶¶$ \left\{{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) $ \left\{\begin{array}{ll} -\mathrm{div}(|\nabla u|^{p-2}\nabla u) + \lambda(x)|u|^{p-2}u = \alpha(x)f(u) + \beta(x)g(u) &; $ \mbox{in $ \mbox{in \Omega$}\\ {\partial u \over \partial \nu} = 0 $}\\ {\partial u \over \partial \nu} = 0 &; $ \mbox{on $ \mbox{on \partial \Omega$} \right. $}\end{array} \right. ¶¶ where $ \nu $ \nu is the outward unit normal to the boundary $ \partial\Omega $ \partial\Omega of the bounded open set _boxclose^N \Omega \subset \mathbb{R}^N . The existence of solutions, for the above problem, is proved by applying a critical point theorem recently obtained by B. Ricceri as a consequence of a more general variational principle.     

Local bifurcation-branching analysis of global and ��blow-up�� patterns for a fourth-order thin film equation

Countable families of global-in-time and blow-up similarity sign-changing patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4)

Solutions of a system of diffusion equations

We study existence and multiplicity of homoclinic type solutions to the following system of diffusion equations on \mathbbR ×W{\mathbb{R}} \times \Omega :

Bifurcation problems for a class of degenerate quasilinear elliptic equations

In this paper we consider the bifurcation problem -div A（x, u）=λa（x）｜u｜^p-2u＋f（x,u,λ） in Ω with p 〉 1.Under some proper assumptions on A（x,ξ）,a（x） and f（x,u,λ）,we show that the existence of an unbounded branch of positive solutions bifurcating Irom the principal eigenvalue of the problem --div A（x, u）=λa（x）｜u｜^p-2u.     

New Kamenev-type oscillation criteria for half-linear partial differential equations

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping under quite general conditions. These results are extensions of the recent results developed by Sun [Y.G. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.     

一类含测度的非强制拟线性椭圆方程解的存在性和不存在性

本文主要研究如下含非线性梯度项的非强制拟线性椭圆方程\begin{equation*}\left \{\begin{array}{rl}-\text{div}(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}})+\frac{|u|^{p-2}u|\nabla u|^{p}}{(1+|u|)^{\theta p}}=\mu,~&x\in\Omega,\\ u=0,~&x\in\partial\Omega,\end{array}\right.\end{equation*} 弱解的存在性和不存在性, 其中$\Omega\subseteq\mathbb{R}^N(N\geq3)$ 是有界光滑区域, $1相似文献   

Blow-up,quenching, aggregation and collapse in a chemotaxis model with reproduction term

In this paper,we consider the following chemotaxis model with ratio-dependent logistic reaction term u/t=D▽(▽u-u▽ω/ω)+u(α-bu/ω),(x,t)∈QT,ω/t=βu-δω,(x,t)∈QT,u▽㏑(u/w)·=0,x ∈Ω,0tT,u(x,0)=u0(x)0,x ∈,w(x,0)=w0(x)0,x ∈,It is shown that the solution to the problem exists globally if b+β≥0 and will blow up or quench if b+β0 by means of function transformation and comparison method.Various asymptotic behavior related to different coefficients and initial data is also discussed.     

Soliton Solutions for a Generalized Quasilinear Elliptic Problem

Potential Analysis - We establish existence and multiplicity of solutions for the elliptic quasilinear Schrödinger equation $$ -\text{div}(g^{2}(u)\nabla u) +g(u)g^{\prime}(u)|\nabla u|^{2}+...     

Regularity Results for a Nonlinear Elliptic-Parabolic System with Oscillating Coefficients

In this paper we study the initial boundary value problem for the system $\mbox{div}(\sigma(u)\nabla\varphi)=0,$ $ u_t-\Delta u=\sigma(u)|\nabla\varphi|^2$. This problem is known as the thermistor problem which models the electrical heating of conductors. Our assumptions on $\sigma(u)$ leave open the possibility that $\liminf_{u\rightarrow\infty}\sigma(u)=0$, while $\limsup_{u\rightarrow\infty}\sigma(u)$ is large. This means that $\sigma(u)$ can oscillate wildly between $0$ and a large positive number as $u\rightarrow \infty$. Thus our degeneracy is fundamentally different from the one that is present in porous medium type of equations. We obtain a weak solution $(u, \varphi)$ with $|\nabla \varphi|, |\nabla u|\in L^\infty$ by first establishing a uniform upper bound for $e^{\varepsilon u}$ for some small $\varepsilon$. This leads to an inequality in $\nabla\varphi$, from which the regularity result follows. This approach enables us to avoid first proving the Hölder continuity of $\varphi$ in the space variables, which would have required that the elliptic coefficient $\sigma(u)$ be an $A_2$ weight. As it is known, the latter implies that $\ln\sigma(u)$ is "nearly bounded''.     

Unique continuation for the system of elasticity in the plane

We prove the strong unique continuation property for the Lamé system of elastostatics in the plane, , with variable Lamé coefficients , , when is Lipschitz and is measurable.

     

Capacity solution for a perturbed nonlinear coupled system

We shall give the existence of a capacity solution to a nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$\varphi $$, the model problem we refer to is $$\begin{aligned} \left\{ \begin{array}{l} \Delta _p u+g(x,u)= \rho (u)|\nabla \varphi |^2 \quad \mathrm{in} \quad \Omega ,\\ {{\,\mathrm{div}\,}}(\rho (u)\nabla \varphi ) =0 \quad \mathrm{in} \quad \Omega ,\\ \varphi =\varphi _0 \quad \text{ on } \quad {\partial \Omega },\\ u=0 \quad \mathrm{on} \quad {\partial \Omega }, \end{array} \right. \end{aligned}$$where $$\Omega \subset \mathbb {R}^N$$, $$N\ge 2$$ and $$\Delta _p u=-{\text {div}}\left( |\nabla u|^{p-2} \nabla u\right) $$ is the so-called p-Laplacian operator, and g a nonlinearity which satisfies the sign condition but without any restriction on its growth. This problem may be regarded as a generalization of the so-called thermistor problem, where we consider the case of the elliptic equation is non-uniformly elliptic.     

A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier–Stokes System involving Dirichlet Boundary Conditions for the Signal

The chemotaxis-Navier-Stokes system ■is considered in a smoothly bounded planar domain Ω under the boundary conditions■ with a given nonnegative constant c_*.It is shown that if(n₀,c₀,u₀) is sufficiently regular and such that the product ■is suitably small,an associated initial value problem possesses a bounded classical solution with(n,c,u)|_t=0=(n₀,c₀,u₀).     

On periodic solutions of non-autonomous second order hamiltonian systems

The purpose of this paper is to study the existence of periodic solutions for the non-autonomous second order Hamiltonian system

Continuous solutions of the first boundary value problems for nonlinear diffusion equations

We establish the existence of continous solutions of the first boundary value problem for nonlinear diffusion equations of the form

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)