Existence of multiple solutions and estimates of extremal values for a Kirchhoff type problem with fast increasing weight and critical nonlinearity
In this article, we study the Kirchhoff type problem $$ -\Big(a+\epsilon\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\Big)\hbox{div} (K(x)\nabla u)=\lambda K(x)f(x)|u|^{q-2}u+K(x)|u|^{4}u, $$ where $x\in \mathbb{R}^3$, $1<q<2$, $K(x)=\exp({|x|^{\alpha}/4})$ with $\alpha\geq2$, $\epsilon>0$...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2018-07-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2018/144/abstr.html |
| Summary: | In this article, we study the Kirchhoff type problem
$$
-\Big(a+\epsilon\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\Big)\hbox{div}
(K(x)\nabla u)=\lambda K(x)f(x)|u|^{q-2}u+K(x)|u|^{4}u,
$$
where $x\in \mathbb{R}^3$, $1<q<2$, $K(x)=\exp({|x|^{\alpha}/4})$ with
$\alpha\geq2$, $\epsilon>0$ is small enough, and the parameters $a, \lambda >0$.
Under some assumptions on $f(x)$, we establish the existence of two nonnegative
nontrivial solutions and obtain uniform lower estimates for extremal values
of the problem via variational methods. |
|---|---|
| ISSN: | 1072-6691 |