Sign-changing solutions for a class of <em>p</em>-Laplacian Kirchhoff-type problem with logarithmic nonlinearity

Ya-Lei Li; Da-Bin Wang; Jin-Long Zhang; Ya-Lei Li; Da-Bin Wang; Jin-Long Zhang

doi:10.3934/math.2020139

AIMS Mathematics

2020, Volume 5, Issue 3: 2100-2112. doi: 10.3934/math.2020139

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Research article

Sign-changing solutions for a class of p-Laplacian Kirchhoff-type problem with logarithmic nonlinearity

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, P. R. China

Received: 19 December 2019 Accepted: 12 February 2020 Published: 26 February 2020
MSC : 35J20, 35J65

In this paper, we study the existence of ground state sign-changing solutions for following $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity $ \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -(a+ b\int _{\Omega}|\nabla u|^{p}dx)\Delta_p u = |u|^{q-2}u\ln u^2, ~x\in\Omega \\ u = 0, ~\ x\in \partial\Omega, \end{array} \right. \end{equation*} $ where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $a, b \gt 0$ are constant, $4\leq 2 p \lt q \lt p^*$ and $N \gt p$. By using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains.
- p-Laplacian Kirchhoff-type equation,
- nonlocal term,
- variation methods,
- sign-changing solutions
Citation: Ya-Lei Li, Da-Bin Wang, Jin-Long Zhang. Sign-changing solutions for a class of p-Laplacian Kirchhoff-type problem with logarithmic nonlinearity[J]. AIMS Mathematics, 2020, 5(3): 2100-2112. doi: 10.3934/math.2020139

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Abstract

In this paper, we study the existence of ground state sign-changing solutions for following $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity $ \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -(a+ b\int _{\Omega}|\nabla u|^{p}dx)\Delta_p u = |u|^{q-2}u\ln u^2, ~x\in\Omega \\ u = 0, ~\ x\in \partial\Omega, \end{array} \right. \end{equation*} $ where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $a, b \gt 0$ are constant, $4\leq 2 p \lt q \lt p^*$ and $N \gt p$. By using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains.

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