Electron. J. Differential Equations, Vol. 2017 (2017), No. 229, pp. 1-16.

Sign-changing solutions for elliptic equations with fast increasing weight and concave-convex nonlinearities

Xiaotao Qian, Jianqing Chen

Abstract:
In this article, we study the problem
$$ -\hbox{div}(K(x)\nabla u)=a(x)K(x)|u|^{q-2}u+b(x)K(x)|u|^{2^{\ast}-2}u, \quad x\in \mathbb{R}^N, $$
where $2^{\ast}=2N/(N-2)$, $N\geq3$, $1<q<2$, $K(x)=\exp({|x|^{\alpha}/4})$ with $\alpha\geq2$. Under some assumptions on the potentials a(x) and b(x), we obtain a pair of sign-changing solutions of the problem via variational methods and certain estimates.

Submitted July 13, 2017. Published September 22, 2017.
Math Subject Classifications: 35J20, 35J60.
Key Words: Sign-changing solutions; variational method; critical problem.

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Xiaotao Qian
College of Mathematics and Computer Science & FJKLMAA
Fujian Normal University
Qishan Campus, Fuzhou 350108, China
email: qianxiaotao1984@163.com
Jianqing Chen
College of Mathematics and Computer Science & FJKLMAA
Fujian Normal University
Qishan Campus, Fuzhou 350108, China
email: jqchen@fjnu.edu.cn

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