发表机构
University of Messina（墨西拿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过变分方法和两个三临界点定理，证明了全空间中 $p>N$ 的 $p$-Laplacian 方程在适当条件下至少存在三个弱解，并给出了参数范围及非负性条件。
AI 中文摘要
本文研究全空间 $\mathbb{R}^N$ 中涉及 $p$-Laplacian 算子（其中 $p>N$）的非线性椭圆问题。在势项和非线性项满足适当假设的条件下，我们证明了至少存在三个不同的弱解。我们的方法是变分的，依赖于两个不同的三临界点定理。这些结果将一些近期关于 $p$-Laplacian 方程的多重性定理推广到全空间，并为参数 $\lambda > 0$ 提供了确保多个解存在的显式范围，在附加符号条件下这些解是非负的。我们还讨论了解的非平凡性。最后，给出了对具有分离变量的非线性问题的应用。
英文摘要
In this paper we deal with a nonlinear elliptic problem in the whole space $\mathbb{R}^N$ involving the $p$-Laplacian operator with $p>N$. Under suitable assumptions on the potential term and on the nonlinearity, we establish the existence of at least three distinct weak solutions. Our approach is variational and relies on two different three critical points theorems. The results extend to higher dimensions some recent multiplicity theorems for one-dimensional $p$-Laplacian equations on the real line, and provide explicit ranges for the parameter $λ> 0$ ensuring the existence of multiple solutions, which are nonnegative under additional sign conditions. The nontriviality of the solutions is also discussed. Finally, applications to problems whose nonlinearities have separated variables are presented.
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