Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities

Yang, Jie; Chen, Haibo; Feng, Zhaosheng

doi:10.58997/ejde.2020.101

ejde

2020

DOI: 10.58997/ejde.2020.101

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Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities

Jie Yang,

Haibo Chen,

Zhaosheng Feng

Abstract: This article concerns the existence and multiplicity of positive solutions to the fractional Kirchhoff equation with critical indefinite nonlinearities by applying the Nehari manifold approach and fibering maps.

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2024

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Cited by 4 publications

References 26 publications

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Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions

Yang,

Liu,

Chen

2024

MATH

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<abstract>This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $ (-\Delta)^{s} $ <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} \left(1+\alpha\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s} u+\mu K(x)u = g(x)|u|^{p-2}u, &{\rm in}\ \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \ \end{cases} $\end{document} </tex-math></disp-formula> where $ \alpha, \ \mu > 0 $, $ s\in [\frac{3}{4}, 1) $, $ 2 </abstract>

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Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions

Yang,

Liu,

Chen

2024

MATH

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Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent

Lin,

Zheng

2021

ejde

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Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain with smooth boundary and $0\in\Omega$. For $0<s<1$, $1\le r<q<p$, $0\le\alpha<ps<N$ and a positive parameter $\lambda$, we consider the fractional $(p,q)$-Laplacian problems involving a critical Sobolev-Hardy exponent. This model comes from a nonlocal problem of Kirchhoff type $$\displaylines{ \big(a+b[u]_{s,p}^{(\theta-1)p}\big)(-\Delta)_{p}^{s}u+(-\Delta)_{q}^{s}u =\frac{|u|^{p_{s}^{*}(\alpha)-2}u}{|x|^{\alpha}}+\lambda f(x)\frac{|u|^{r-2}u}{|x|^{c}}\quad \hbox{in }\Omega,\cr u=0\quad\text{in }\mathbb{R}^{N}\setminus\Omega, }$$ where $a,b>0$, $c<sr+N(1-r/p)$, $\theta\in(1,p_{s}^{*}(\alpha)/p)$ and $p_{s}^{*}(\alpha)$ is critical Sobolev-Hardy exponent. For a given suitable $f(x)$, we prove that there are least two nontrivial solutions for small $\lambda$, by way of the mountain pass theorem and Ekeland's variational principle. Furthermore, we prove that these two solutions converge to two solutions of the limiting problem as $a\to 0^{+}$. For the limiting problem, we show the existence of infinitely many solutions, and the sequence tends to zero when $\lambda$ belongs to a suitable range. For more information see https://ejde.math.txstate.edu/Volumes/2021/66/abstr.html

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Periodic solutions for conformable type non-instantaneous impulsive differential equations

Ding,

Wang

2021

ejde

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In this article we study a type of conformable non-instantaneous impulsive equation with periodic effects. We find a Cauchy matrix that can provide solutions of linear and nonlinear problems and prove some of their properties. Also we study the existence of periodic solution of different types of conformable non-instantaneous impulsive differential equation. Some examples also are given to illustrate our theoretical results. For more information see https://ejde.math.txstate.edu/Volumes/2021/94/abstr.html

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Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities

Abstract: This article concerns the existence and multiplicity of positive solutions to the fractional Kirchhoff equation with critical indefinite nonlinearities by applying the Nehari manifold approach and fibering maps.

Cited by 4 publications

References 26 publications

Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions

Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions

Multiplicity and asymptotic behavior of solutions to fractional (p,q)-Kirchhoff type problems with critical Sobolev-Hardy exponent

Periodic solutions for conformable type non-instantaneous impulsive differential equations

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