October 12, 2025Open Access

Multiple solutions to a class of p-Laplacian Schrödinger equations

Key Points

Infinitely many solutions exist for the p-Laplacian Schrödinger equation, providing new insights into this domain.
The analysis utilizes the variational perturbation method to construct perturbation spaces, expanding traditional approaches.
Previous limitations of discussing solution multiplicity under the Palais-Smale condition in W^{1,p} spaces are addressed effectively.
The findings extend classical results on p-Laplacian equations and answer an open question posed by Candela et al.

Abstract

In this paper, we will prove the existence of infinitely many solutions to the following equation by utilizing the variational perturbation method equation* -div (A (x, u) | u|^p-2 u) +1pAₓ (x, u) | u|^p+V (x) |u|^p-2u=g (x, u), ~~ x^N, equation* where 2<p<N, V (x) represents a coercive potential. It should be emphasized that the multiplicity of solutions is unsuitable to be discussed under the framework of W^1, p (R^N) L^ (R^N), as the functional corresponding to the above equation does not satisfy the Palais-Smale condition in this space. To this end, we will use the variational perturbation method to construct novel perturbation space and perturbation functional. By relying on the established conclusions regarding the multiplicity of solutions for classical p-Laplacian equations, we shall analyze the equation in question. Our result addresses positively the open question from Candela et. al in CVPDE.

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