Existence of solutions for fractional $p$-Kirchhoff equations with critical nonlinearities
In this article, we show the existence of non-negative solutions of the fractional p-Kirchhoff problem $$\displaylines{ -M(\int_{\mathbb{R}^{2n}} |u(x)-u(y)|^pK(x-y)dx\,dy)\mathcal{L}_Ku =\lambda f(x,u)+|u|^{p^* -2}u\quad \text{in }\Omega,\cr u=0\quad \text{in }\mathbb{R}^{n}\setminus\Omega, }...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-04-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/93/abstr.html |
| Summary: | In this article, we show the existence of non-negative solutions of the
fractional p-Kirchhoff problem
$$\displaylines{
-M(\int_{\mathbb{R}^{2n}} |u(x)-u(y)|^pK(x-y)dx\,dy)\mathcal{L}_Ku
=\lambda f(x,u)+|u|^{p^* -2}u\quad \text{in }\Omega,\cr
u=0\quad \text{in }\mathbb{R}^{n}\setminus\Omega,
}$$
where $\mathcal{L}_K$ is a p-fractional type non local operator with kernel K,
$\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary,
M and f are continuous functions, and $p^*$ is the fractional Sobolev exponent. |
|---|---|
| ISSN: | 1072-6691 |