Search results for "Galerkin basis"
showing 2 items of 2 documents
Variable exponent p(x)-Kirchhoff type problem with convection
2022
Abstract We study a nonlinear p ( x ) -Kirchhoff type problem with Dirichlet boundary condition, in the case of a reaction term depending also on the gradient (convection). Using a topological approach based on the Galerkin method, we discuss the existence of two notions of solutions: strong generalized solution and weak solution. Strengthening the bound on the Kirchhoff type term (positivity condition), we establish existence of weak solution, this time using the theory of operators of monotone type.
The existence of solutions for the modified (p(x), q(x))-Kirchhoff equation
2022
We consider the Dirichlet problem-Delta(Kp)(p(x))u(x) - Delta(Kq)(q(x))u(x) = f(x, u(x), del u(x)) in Omega, u vertical bar(partial derivative Omega) = 0,driven by the sum of a p(x)-Laplacian operator and of a q(x)-Laplacian operator, both of them weighted by indefinite (sign-changing) Kirchhoff type terms. We establish the existence of weak solution and strong generalized solution, using topological tools (properties of Galerkin basis and of Nemitsky map). In the particular case of a positive Kirchhoff term, we obtain the existence of weak solution (= strong generalized solution), using the properties of pseudomonotone operators.