We study heat equtions with generalized fractional Laplacian, which is defined by the spectral theory. Here, we develop the existence theory for that equations. Moreover, we present some numerical simulations for our problems.

We provide parameter dependent version of the Browder–Minty Theorem in case when the solution is unique utilizing different types of monotonicity and compactness assumptions related to condition (S)2. Potential equations and the convergence of their Euler action functionals is also investigated. Applications towards the dependence on parameters for both potential and non-potenial nonlinear Dirichlet boundary problems are given.