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 divΦ(x,u)and a gradient term H(x,u,▽u)with no growth r...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 divΦ(x,u)and a gradient term H(x,u,▽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{▽u^(ε)}_(ε),which helps to pass to the limit to find a weak solution.展开更多
We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)...We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)lie in the space Lr(0,T;Lq(Ω)),γis a positive constant.The purpose of this paper is to prove,under suitable assumptions on the integrability of the space Lr(0,T;Lq(Ω))for the source terms and the coefficient of the gradient term,a priori L^(∞)estimate and the existence of bounded solutions.The methods consist of constructing a family of perturbation problems by regularization,Stampacchia’s iterative technique fulfilled by an appropriate nonlinear test function and compactness argument for the limit process.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11901131)the University-Level Research Fund Project in Guizhou University of Finance and Economics(Grant No.2022KYYB01)。
文摘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 divΦ(x,u)and a gradient term H(x,u,▽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{▽u^(ε)}_(ε),which helps to pass to the limit to find a weak solution.
基金Supported by the National Natural Science Foundation of China(Grant No.11901131)the University-Level Research Fund Project in Guizhou University of Finance and Economics(Grant No.2019XYB08)。
文摘We consider a class of nonlinear parabolic equations whose prototype is ut-Δu=b(x,t)·■+γ|■u|^(2)-divF(x,t)+f(x,t),(x,t)∈ΩT,u(x,t)=∈ГT,u(x,0)=u0(x),x∈Ω where the functions|b(x,t)|^(2),|F(x,t)|^(2),f(x,t)lie in the space Lr(0,T;Lq(Ω)),γis a positive constant.The purpose of this paper is to prove,under suitable assumptions on the integrability of the space Lr(0,T;Lq(Ω))for the source terms and the coefficient of the gradient term,a priori L^(∞)estimate and the existence of bounded solutions.The methods consist of constructing a family of perturbation problems by regularization,Stampacchia’s iterative technique fulfilled by an appropriate nonlinear test function and compactness argument for the limit process.
