Communications in Mathematical Analysis

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

Duong Trong Luyen

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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

Permanent link to this document
https://projecteuclid.org/euclid.cma/1566266428

Mathematical Reviews number (MathSciNet)
MR3992902

Subjects
Primary: 35J70: Degenerate elliptic equations 35J20: Variational methods for second-order elliptic equations
Secondary: 35J10: Schrödinger operator [See also 35Pxx]

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

Citation

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


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