一类具低阶项和变指数的椭圆方程解的存在性

Existence of Solution to a Class of Elliptic Equations with Lower Order Terms and Variable Exponents

中文关键词:  椭圆方程  非标准增长条件  低阶项  弱解

英文关键词:elliptic equations  nonstandard growth condition  lower order terms  weak solutions

中文摘要:

      研究一类具非标准增长条件的椭圆方程. 方程具变指数且非强制的散度项div$\phi(x,u)$和对$u$无增长限制的梯度项$H(x,u,\nabla u)$同时出现, 这些特点使得无法直接使用经典的存在性理论得到存在性. 在扰动问题中选取合适的检验函数, 得到了一些在变指数框架下关于解的先验估计. 基于这些估计, 我们证明了梯度序列$\{\nabla u^\epsilon\}_\epsilon$的几乎处处收敛, 进一步通过取极限得到了弱解的存在性.

英文摘要:

      We study a class of nonlinear elliptic equations with nonstandard growth condition. The main feature is that two lower order terms, a non-coercive divergence term $\text{div}\Phi(x,u)$ and a gradient term $H(x,u,\nabla u)$ with no growth restriction on $u$, appear simultaneously in the variable exponents setting. These characteristics prevent us from directly obtaining the existence of solutions by employing the classical theory on existence results. By choosing some appropriate test functions in the perturbed problem, some a priori estimates are obtained under the variable exponent framework. Based on these estimates, we prove the almost everywhere convergence of the gradient sequence $\{\nabla u^\epsilon\}_\epsilon$, which helps to pass to the limit to find a weak solution.