A new approach to solving a quasilinear boundary value problem with p-Laplacian using optimization

We present a novel approach to solving a specific type of quasilinear boundary value problem with p-Laplacian that can be considered an alternative to the classic approach based on the mountain pass theorem. We introduce a new way of proving the existence of nontrivial weak solutions. We show that the nontrivial solutions of the problem are related to critical points of a certain functional different from the energy functional, and some solutions correspond to its minimum. This idea is new even for p = 2. We present an algorithm based on the introduced theory and apply it to the given problem. The algorithm is illustrated by numerical experiments and compared with the classic approach.