A new augmented singular transform and its partial Newton-correction method for finding more solutions to nonvariational quasilinear elliptic PDEs

Abstract In this paper, in order to find more solutions to a nonvariational quasilinear PDE, a new augmented singular transform (AST) is developed to form a barrier surrounding previously found solutions so that an algorithm search from outside cannot pass the barrier and penetrate into the inside to reach a previously found solution. Thus a solution found by the algorithm must be new. Mathematical justifications of AST formulation are established. A partial Newton-correction method is designed accordingly to solve the augmented problem and to satisfy a constraint in AST. The new method is applied to numerically investigate bifurcation, symmetry-breaking phenomena to a non-variational quasilinear elliptic equation through finding multiple solutions. Such phenomena are numerically captured and visualized for the first time, and still open for theoretical verification. Since the formulation is general and simple, it opens a door to solve other multiple solution problems.

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