Electron. J. Differential Equations, Vol. 2017 (2017), No. 180, pp. 1-15.

Sign-changing solutions for non-local elliptic equations

Huxiao Luo

Abstract:
This article concerns the existence of sign-changing solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions,
$$\displaylines{ -\mathcal{L}_Ku=f(x,u),\quad x\in \Omega, \cr u=0,\quad x\in \mathbb{R}^n\setminus\Omega, }$$
where $\Omega\subset\mathbb{R}^n\; (n\geq2)$ is a bounded, smooth domain and the nonlinear term f satisfies suitable growth assumptions. By using Brouwer's degree theory and Deformation Lemma and arguing as in [2], we prove that there exists a least energy sign-changing solution. Our results generalize and improve some results obtained in [27]

Submitted March 30, 2017. Published July 14, 2017.
Math Subject Classifications: 35R11, 58E30.
Key Words: Brouwer's degree theory; sign-changing solutions; non-local elliptic equations; deformation Lemma.

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Huxiao Luo
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: luohuxiao1989@163.com

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