English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The purpose of this paper is to study the indefinite Kirchhoff type problem:\begin{equation*} \left\{ \begin{array}{ll} M\left(\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right) \left[ -\Delta u+u\right]=f(x,u) & \text{in }\mathbb{R}^{N}, \\ 0\leq u\in H^{1}\left(\mathbb{R}^{N}\right), & \end{array} \right. \end{equation*} where $N\geq 1$,$M(t)=am\left( t\right) +b$, $m\in C(\mathbb{R}^{+})$ and $f(x,u)=g(x,u)+h(x)u^{q-1}$. We require that $f$ is \textquotedblleftlocal\textquotedblright\ sublinear at the origin and \textquotedblleftlocal\textquotedblright\ linear at infinite. Using the mountain pass theoremand Ekeland variational principle, the existence and multiplicity of nontrivialsolutions are obtained. In particular, the criterion of existence of threenontrivial solutions is established.

On the indefinite Kirchhoff type problems with local sublinearity and linearity