Taiwanese Journal of Mathematics

Multiple Solutions of Nonlinear Schrödinger Equations with the Fractional $p$-Laplacian

Huxiao Luo, Xianhua Tang, and Shengjun Li

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We use two variant fountain theorems to prove the existence of infinitely many weak solutions for the following fractional $p$-Laplace equation\[  (-\Delta)^\alpha_p u + V(x) |u|^{p-2}u  = f(x,u), \quad x \in \mathbb{R}^N,\]where $N \geq 2$, $p \geq 2$, $\alpha \in (0,1)$, $(-\Delta)^\alpha_p$ is the fractional $p$-Laplacian and $f$ is either asymptotically linear or subcritical $p$-superlinear growth. Under appropriate assumptions on $V$ and $f$, we prove the existence of infinitely many nontrivial high or small energy solutions. Our results generalize and extend some existing results.

Article information

Taiwanese J. Math., Volume 21, Number 5 (2017), 1017-1035.

Received: 22 September 2016
Revised: 20 December 2016
Accepted: 4 January 2017
First available in Project Euclid: 1 August 2017

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Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 35A15: Variational methods 58E30: Variational principles

fractional $p$-Laplacian nonlinear Schrödinger equation infinitely many solutions fountain theorem


Luo, Huxiao; Tang, Xianhua; Li, Shengjun. Multiple Solutions of Nonlinear Schrödinger Equations with the Fractional $p$-Laplacian. Taiwanese J. Math. 21 (2017), no. 5, 1017--1035. doi:10.11650/tjm/7947. https://projecteuclid.org/euclid.twjm/1501599181

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