Abstract We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form Hölder spaces with variable exponent. The procedure allows extend the up portion boundary where Dirichlet or Neumann conditions are prescribed produces Schauder theory partial derivatives solutions any $$k\in {\mathbb {N...

We study a nonlinear p(x)-Kirchhoff type problem with Dirichlet boundary condition, in the case of reaction term depending also on gradient (convection). Using topological approach based Galerkin method, we discuss existence two notions solutions: strong generalized solution and weak solution. Strengthening bound Kirchhoff (positivity condition), establish solution, this time using theory opera...