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2019 Multiple Solutions for Semilinear $\Delta_{\gamma}-$differential Equations in $\mathbb R^N$ with Sign-changing Potential
Duong Trong Luyen
Commun. Math. Anal. 22(1): 61-75 (2019).

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.

Citation

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Duong Trong Luyen. "Multiple Solutions for Semilinear $\Delta_{\gamma}-$differential Equations in $\mathbb R^N$ with Sign-changing Potential." Commun. Math. Anal. 22 (1) 61 - 75, 2019.

Information

Published: 2019
First available in Project Euclid: 20 August 2019

zbMATH: 07161356
MathSciNet: MR3992902

Subjects:
Primary: 35J20 , 35J70
Secondary: 35J10

Keywords: $\Delta_\gamma-$Laplace problems , multiple solutions , sign-changing potential , variational method , weak solutions

Rights: Copyright © 2019 Mathematical Research Publishers

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Vol.22 • No. 1 • 2019
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