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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper, we study the multiplicity of solutions for the following nonhomogeneous p-Kirchhoff elliptic equation: 0.1(a+λ(∫RN(|∇u|p+|u|p)dx)m)(−Δpu+|u|p−2u)=f(u)+h(x),x∈RN,$$ \biggl(a+\lambda \biggl( \int_{\mathbb{R}^{N}} \bigl(|\nabla{u}|^{p}+|u|^{p} \bigr)\,dx \biggr)^{m} \biggr) \bigl(-\Delta_{p}u+|u|^{p-2}u \bigr) =f(u)+h(x),\quad x\in\mathbb{R}^{N}, $$ with a,λ,m>0$a,\lambda,m>0$ and 1<p<N$1< p< N$. By variational methods we prove that problem (0.1) admits at least two solutions under appropriate assumptions on f(u)$f(u)$ and h(x)$h(x)$. The main difficulty to overcome is the lack of an a priori bound for Palais-Smale sequence. Motivated by Jeanjean (Proc. R. Soc. Edinb., Sect. A 129:787-809, 1999), we use a cut-off functional to obtain a bounded (PS) sequence. Also, if f(u)=|u|q−2u$f(u)=|u|^{q-2}u$, p<q<min{p(m+1),p∗=pNN−p}$p< q<\min\{p(m+1), p^{*}=\frac{pN}{N-p}\}$, and h(x)=0$h(x)=0$, then we prove that problem (0.1) has at least one nontrivial solution for any λ∈(0,λ∗]$\lambda\in(0, \lambda^{*}]$ and has no nontrivial weak solutions for any λ∈(λ∗,+∞)$\lambda\in(\lambda^{*}, +\infty)$.

Existence,multiplicity, and nonexistence of solutions for a p-Kirchhoff elliptic equation on R N $\mathbb{R}^{N}$