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$p$-Laplacian operator; quasilinear elliptic PDE; critical point and value; optimization algorithm; gradient method
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.
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