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

Abstract:

      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.