## Communications in Mathematical Analysis

### Multiple Solutions for Semilinear $\Delta_{\gamma}-$differential Equations in $\mathbb R^N$ with Sign-changing Potential

Duong Trong Luyen

#### Abstract

In this paper, we study the existence of infinitely many nontrivial solutions of the semilinear $\Delta_{\gamma}$ differential equations in $\mathbb{R}^N$ $$- \Delta_{\gamma}u+ b(x)u=f(x,u)\quad \mbox{ in }\; \mathbb{R}^N, \quad u \in S^2_{\gamma}(\mathbb{R}^N),$$ where $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \quad \partial_{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma = (\gamma_1, \gamma_2, ..., \gamma_N),$$ and the potential $b$ is allowed to be sign-changing, and the primitive of the nonlinearity $f$ is of superquadratic growth near infinity in $u$ and allowed to be sign-changing.

#### Article information

Source
Commun. Math. Anal., Volume 22, Number 1 (2019), 61-75.

Dates
First available in Project Euclid: 20 August 2019

Luyen, Duong Trong. Multiple Solutions for Semilinear $\Delta_{\gamma}-$differential Equations in $\mathbb R^N$ with Sign-changing Potential. Commun. Math. Anal. 22 (2019), no. 1, 61--75. https://projecteuclid.org/euclid.cma/1566266428