POSITIVE SOLUTIONS AND BIFURCATION OF FULLY NONLINEAR ELLIPTIC EQUATIONS INVOLVING SUPER－CRITICAL SOBOLEV EXPONENTS

ＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＡＮＤＢＩＦＵＲＣＡＴＩＯＮＯＦＦＵＬＬＹＮＯＮＬＩＮＥＡＲＥＬＬＩＰＴＩＣＥＱＵＡＴＩＯＮＳＩＮＶＯＬＶＩＮＧＳＵＰＥＲ－ＣＲＩＴＩＣＡＬＳＯＢＯＬＥＶＥＸＰＯＮＥＮＴＳ￥ＱＵＣＨＡＮＧＺＨＥＮＧ（屈长征）（Ｉｎｓ...     

The blow-up property for a system of heat equations with nonlinear boundary conditions

ＴＨＥＢＬＯＷ┐ＵＰＰＲＯＰＥＲＴＹＦＯＲＡＳＹＳＴＥＭＯＦＨＥＡＴＥＱＵＡＴＩＯＮＳＷＩＴＨＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＣＯＮＤＩＴＩＯＮＳＬＩＮＺＨＩＧＵＩ,ＸＩＥＣＨＵＮＨＯＮＧＡＮＤＷＡＮＧＭＩＮＧＸＩＮＡｂｓｔｒａｃｔ．Ｔｈｉｓｐａｐ...     

A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS

ＡＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＯＦＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＳＯＭＥＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺＨＡＮＧＱＵＡＮＤＥ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

AN EXISTENCE－UNIQUENESS RESULT FOR SECOND ORDER NONLINEAR DIFEERENTIAL EQUATIONSWITH ROBIN BOUNDARY CONDITION

ＡＮＥＸＩＳＴＥＮＣＥ－ＵＮＩＱＵＥＮＥＳＳＲＥＳＵＬＴＦＯＲＳＥＣＯＮＤＯＲＤＥＲＮＯＮＬＩＮＥＡＲＤＩＦＥＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷＩＴＨＲＯＢＩＮＢＯＵＮＤＡＲＹＣＯＮＤＩＴＩＯＮ￥ＬｉｎＺｈｅｎｇｈｕａ（ＪｉｌｉｎＵｎｉｖｅｒｓｉｔ...     

NONNULL DISTRIBUTION OF THE LIKELIHOOD RATIO CRITERION FOR TESTING THE EQUALITY OF DIAGONAL BLOCKS WITH BLOCKWISE INDEPENDENCE

ＮＯＮＮＵＬＬＤＩＳＴＲＩＢＵＴＩＯＮＯＦＴＨＥＬＩＫＥＬＩＨＯＯＤＲＡＴＩＯＣＲＩＴＥＲＩＯＮＦＯＲＴＥＳＴＩＮＧＴＨＥＥＱＵＡＬＩＴＹＯＦＤＩＡＧＯＮＡＬＢＬＯＣＫＳＷＩＴＨＢＬＯＣＫＷＩＳＥＩＮＤＥＰＥＮＤＥＮＣＥＡ．Ｋ．Ｇｕｐｔａ；Ｃ．Ｃ．Ｃ...     

THE EXISTENCE AND UNIQUENESS AND STABILITY OF ALMOST PERIODIC SOLUTIONS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

ＴＨＥＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＡＮＤＳＴＡＢＩＬＩＴＹＯＦＡＬＭＯＳＴＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＦＯＲＦＵＮＣＴＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷＩＴＨＩＮＦＩＮＩＴＥＤＥＬＡＹＳＷＡＮＧＱＵＡＮＨＮＹ...     

FREDHOLM DETERMINANT SOLUTION FOR NONLINEAR EQUATIONS ASSOCIATED WITH ISOSPECTRAL SCHRODINGER EICENVALUE PROBLEM

ＦＲＥＤＨＯＬＭＤＥＴＥＲＭＩＮＡＮＴＳＯＬＵＴＩＯＮＦＯＲＮＯＮＬＩＮＥＡＲＥＱＵＡＴＩＯＮＳＡＳＳＯＣＩＡＴＥＤＷＩＴＨＩＳＯＳＰＥＣＴＲＡＬＳＣＨＲＯＤＩＮＧＥＲＥＩＣＥＮＶＡＬＵＥＰＲＯＢＬＥＭＨｕａｎｇＬｉｅｄｅ（黄烈德）；ＸｉａＺｈｏｎｇ...     

SINGULAR PERTURBATION OF ROBIN BOUNDARY VALUE PROBLEM FOR THIRD ORDER NONLINEAR SYSTEM WITH BOUNDARY PERTURBATION

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＯＦＲＯＢＩＮＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＦＯＲＴＨＩＲＤＯＲＤＥＲＮＯＮＬＩＮＥＡＲＳＹＳＴＥＭＷＩＴＨＢＯＵＮＤＡＲＹＰＥＲＴＵＲＢＡＴＩＯＮ（黄晓秋）福建师范大学福清分校，邮编：３５０３００Ｈ...     

EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION

ＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＯＦＴＨＥＥＮＴＲＯＰＹＳＯＬＵＴＩＯＮＴＯＡＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮ￥Ｒ．ＥＹＭＡＲＤ；Ｔ．ＧＡＬＬＯＵＥＴ；Ｒ．ＨＥＲＢＩＮ（ＬａｂｏｒａｔｏｉｒｅＣｅｎｔｒａｌｄｅｓＰｏ...     

STABILITY OF SOLUTIONS OF FIRST ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS WITH OSCILLATORY COEFFICIENTS

ＳＴＡＢＩＬＩＴＹＯＦＳＯＬＵＴＩＯＮＳＯＦＦＩＲＳＴＯＲＤＥＲＮＯＮＬＩＮＥＡＲＮＥＵＴＲＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷＩＴＨＯＳＣＩＬＬＡＴＯＲＹＣＯＥＦＦＩＣＩＥＮＴＳ￥ＷａｎｇＱｉｒｕ(王其如）（ＬｕｏｙａｎｇＴｅａｃｈｅｒ?..     

Chemotaxis-fluild 系统的全局弱解

We investigate the existence of the global weak solution to the coupled Chemotaxisfluid system ■in a bounded smooth domain ??R~2. Here, r≥0 and μ 0 are given constants,?Φ∈L~∞(?) and g∈L~2((0, T); L_σ~2(?)) are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.     

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)      13. 一类椭圆型随机偏微分方程弱解的存在性          下载免费PDF全文 冉启康 《数学物理学报(A辑)》2008,28(2):320-328 设$D$是$R^N$ ($N>1$)中有界开集,$(\Omega, {\cal F}, P)$是一个完备的概率空间.该文研究了下列随机边值问题弱解的存在性问题\[\left\{\begin{array}{ll}-{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &;(x,\omega)\in D\times \Omega,\\u=0, &;(x,\omega)\in \partial D\times \Omega,\end{array}\right.\]其中, div与 $\nabla $ 表示仅对 $x$求微分. 首先,作者引入了弱解的概念; 然后,作者转化随机问题为高维确定性问题;最后,作者证明了该问题弱解的存在性.      14. On a class of singular p-Laplacian semipositone problems with sign-changing weights          下载免费PDF全文 S.H. Rasouli  Z. Firouzjahi 《Journal of Applied Analysis & Computation》2014,4(4):383-388 We study existence of positive weak solution for a class of $p$-Laplacian problem $$\left\{\begin{array}{ll}-\Delta_{p}u = \lambda g(x)[f(u)-\frac{1}{u^{\alpha}}], & x\in \Omega,\\u= 0 , & x\in\partial \Omega,\end{array\right.$$ where $\lambda$ is a positive parameter and $\alpha\in(0,1),$ $\Omega $ is a bounded domain in $ R^{N}$ for $(N > 1)$ with smooth boundary, $\Delta_{p}u = div (|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator for $( p > 2),$ $g(x)$ is $C^{1}$ sign-changing function such that maybe negative near the boundary and be positive in the interior and $f$ is $C^{1}$ nondecreasing function $\lim_{s\to\infty}\frac{f(s)}{s^{p-1}}=0.$ We discuss the existence of positive weak solution when $f$ and $g$ satisfy certain additional conditions. We use the method of sub-supersolution to establish our result.      15. 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      日毛吉  李翔睿  田巧玉  黄水波 《数学研究及应用》2022,42(2):173-188 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    16. An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights          下载免费PDF全文 S. H. Rasouli & H. Norouzi 《偏微分方程(英文版)》2015,28(1):1-8 We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.      17. 一类具有奇异灵敏度的logistic趋化系统解的渐近行为      杜宛娟 《数学研究及应用》2021,41(5):473-480 本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.      18. 一类Kirchhoff型系统解的存在性      刘晓敏  杨作东 《数学研究及应用》2018,38(4):411-417 In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C([0,∞) × [0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.      19. On Global Solvability of Nonlinear Kirchhoff Model in Infinitely Increasing Moving Domains      Rachid Benabidallah  FranÇois Ebobisse 《Applicable analysis》2013,92(3):705-723 The purpose of this article is to study the existence and uniqueness of global solution for the nonlinear hyperbolic-parabolic equation of Kirchhoff-Carrier type: $$ u_{tt} + \mu u_t - M\left (\int _{\Omega _t}|\nabla u|^2dx\right )\Delta u = 0\quad \hbox {in}\ \Omega _t\quad \hbox {and}\quad u|_{\Gamma _t} = \dot \gamma $$ where $ \Omega _t = \{x\in {\shadR}^2 | \ x = y\gamma (t), \ y\in \Omega \} $ with boundary o t , w is a positive constant and n ( t ) is a positive function such that lim t M X n ( t ) = + X . The real function M is such that $ M(r) \geq m_0 \gt 0 \forall r\in [0,\infty [ $ .      20. Entropy solutions to nonlinear elliptic anisotropic problem with variable exponent          下载免费PDF全文 Benboubker Mohamed Badr  Hjiaj Hassan  OUARO Stanislas 《Journal of Applied Analysis & Computation》2014,4(3):245-270 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 $L^1(\Omega)$.

一类椭圆型随机偏微分方程弱解的存在性

设$D$是$R^N$ ($N>1$)中有界开集,$(\Omega, {\cal F}, P)$是一个完备的概率空间.该文研究了下列随机边值问题弱解的存在性问题\[\left\{\begin{array}{ll}-{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &;(x,\omega)\in D\times \Omega,\\u=0, &;(x,\omega)\in \partial D\times \Omega,\end{array}\right.\]其中, div与 $\nabla $ 表示仅对 $x$求微分. 首先,作者引入了弱解的概念; 然后,作者转化随机问题为高维确定性问题;最后,作者证明了该问题弱解的存在性.     

On a class of singular p-Laplacian semipositone problems with sign-changing weights

We study existence of positive weak solution for a class of $p$-Laplacian problem $$\left\{\begin{array}{ll}-\Delta_{p}u = \lambda g(x)[f(u)-\frac{1}{u^{\alpha}}], & x\in \Omega,\\u= 0 , & x\in\partial \Omega,\end{array\right.$$ where $\lambda$ is a positive parameter and $\alpha\in(0,1),$ $\Omega $ is a bounded domain in $ R^{N}$ for $(N > 1)$ with smooth boundary, $\Delta_{p}u = div (|\nabla u|^{p-2}\nabla u)$ is the p-Laplacian operator for $( p > 2),$ $g(x)$ is $C^{1}$ sign-changing function such that maybe negative near the boundary and be positive in the interior and $f$ is $C^{1}$ nondecreasing function $\lim_{s\to\infty}\frac{f(s)}{s^{p-1}}=0.$ We discuss the existence of positive weak solution when $f$ and $g$ satisfy certain additional conditions. We use the method of sub-supersolution to establish our result.     

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   

An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights

We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.     

一类具有奇异灵敏度的logistic趋化系统解的渐近行为

本文在无边界流的光滑有界区域$\Omega\subset\mathbb{R}^n~(n>2)$上研究了具有奇异灵敏度及logistic源的抛物-椭圆趋化系统$$\left\{\begin{array}{ll}u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+r u-\mu u^k,&x\in\Omega,\,t>0,\\ 0=\Delta v-v+u,&x\in\Omega,\,t>0\end{array}\right.$$ 其中$\chi$, $r$, $\mu>0$, $k\geq2$. 证明了若当$r$适当大, 则当$t\rightarrow\infty$时该趋化系统全局有界解呈指数收敛于$((\frac{r}{\mu})^{\frac{1}{k-1}}, (\frac{r}{\mu})^{\frac{1}{k-1}})$.     

一类Kirchhoff型系统解的存在性

In this paper,we are interested in the existence of positive solutions for the Kirchhoff type problems{-(a_1 + b_1M_1(∫_?|▽u|~pdx))△_(_pu) = λf(u,v),in ?,-(a_2 + b_2M_2(∫?|▽v|~qdx))△_(_qv) = λg(u,v),in ?,u = v = 0,on ??,where 1 p,q N,M i:R_0~+→ R~+(i = 1,2) are continuous and increasing functions.λ is a parameter,f,g ∈ C~1((0,∞) ×(0,∞)) × C([0,∞) × [0,∞)) are monotone functions such that f_s,f_t,g_s,g_t ≥ 0,and f(0,0) 0,g(0,0) 0(semipositone).Our proof is based on the sub-and super-solutions techniques.     

On Global Solvability of Nonlinear Kirchhoff Model in Infinitely Increasing Moving Domains

The purpose of this article is to study the existence and uniqueness of global solution for the nonlinear hyperbolic-parabolic equation of Kirchhoff-Carrier type: $$ u_{tt} + \mu u_t - M\left (\int _{\Omega _t}|\nabla u|^2dx\right )\Delta u = 0\quad \hbox {in}\ \Omega _t\quad \hbox {and}\quad u|_{\Gamma _t} = \dot \gamma $$ where $ \Omega _t = \{x\in {\shadR}^2 | \ x = y\gamma (t), \ y\in \Omega \} $ with boundary o t , w is a positive constant and n ( t ) is a positive function such that lim t M X n ( t ) = + X . The real function M is such that $ M(r) \geq m_0 \gt 0 \forall r\in [0,\infty [ $ .     

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 $L^1(\Omega)$.