A multiplicity result for a class of fractional p-Laplacian equations with perturbations in ℝ^N

ABSTRACT

This paper deals with a class of nonlinear elliptic equations with perturbations in the whole space involving the fractional p-Laplacian. As a particular case, we investigate the following Schrödinger equations with perturbations: $(-\mathrm{\Delta }{)}_p^{s}u+V(x)u=g(x\left)f\right(u)+h(x)x\in {\mathbb{R}}^N,$ where $(-\mathrm{\Delta }{)}_p^s$ is the fractional p-Laplacian operator, $V\left(x\right)$ is a positive continuous function, $h\left(x\right)$ is a perturbation. We first establish a compactness theorem which allows us to give some estimates of the energy levels where the Palais-Smale condition can fail. Furthermore, using Ekeland's variational principle and the mountain pass theorem, we obtain the existence of at least two distinct nonnegative weak solutions for the above-mentioned equations.

COMMUNICATED BY:

• V. Radulescu

KEYWORDS:

• Fractional Laplacian
• perturbation
• Ekeland's variational principle
• mountain pass theorem

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35A15
• 35J60
• 46E35

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

X. Zhang was supported by the National Science Foundation of China (No. 11601103, No. 11671111). B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199).