On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms

Abstract

We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: \begin{equation*} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+1\right) Δu+μV(x)u=λf(x)u+g(x)|u|^{p-2}u\quad \text{ in }\mathbb{R}^{N}, \end{equation*}% where $N\ge 3,2<p<2^{\ast }:=\frac{2N}{N-2}$, $V\in C({\mathbb{R}}^N)$ is a potential well with the bottom $\Omega :=int\{x\in {\mathbb{R}}^N|V(x)=0\}$. When $N=3$ and $4<p<6$, for each $a>0$ and $\mu$ sufficiently large, we obtain that at least one positive solution exists for while at least two positive solutions exist for ${\lambda }_1(f_{\Omega })<\lambda <{\lambda }_1(f_{\Omega })+{\delta }_a$ without any assumption on the integral , where ${\lambda }_1(f_{\Omega })>0$ is the principal eigenvalue of $-\Delta$ in $H_0^1(\Omega )$ with weight function $f_{\Omega }:=f{|}_{\Omega }$, and ${\varphi }_1>0$ is the corresponding principal eigenfunction. When $N\ge 3$ and $2<p<\min \{4,2^{\ast }\}$, for sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<\lambda <{\lambda }_1(f_{\Omega })$; under the classical assumption ${\int }_{\Omega }g(x){\varphi }_1^{p}dx<0$, at least three positive solutions exist for $a>0$ small and ${\lambda }_1(f_{\Omega })\le \lambda <{\lambda }_1(f_{\Omega })+\overline{\delta }$; $(iii)$ under the assumption ${\int }_{\Omega }g(x){\varphi }_1^{p}dx>0$, at least two positive solutions exist for $a>a_0(p)$ and ${\lambda }_a^{+}<\lambda <{\lambda }_1(f_{\Omega })$ for some $a_0(p)>0$ and ${\lambda }_a^{+}\ge 0$.

References in corpus (1)

• On non-local nonlinear elliptic equations involving an eigenvalue problem