On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms
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| Formato: | Preprint |
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2020
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| _version_ | 1866909226200203264 |
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| author | Sun, Juntao Wang, Kuan-Hsiang Wu, Tsung-fang |
| author_facet | Sun, Juntao Wang, Kuan-Hsiang Wu, Tsung-fang |
| contents | 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 $N\geq 3,2<p<2^{\ast }:=\frac{2N}{N-2}$, $V\in C(\mathbb{R}^{N})$ is a potential well with the bottom $Ω:=int\{x\in \mathbb{R}^{N}\ |\ V(x)=0\}$. When $N=3$ and $4<p<6$, for each $a>0$ and $μ$ sufficiently large, we obtain that at least one positive solution exists for $% 0<λ\leqλ_{1}(f_Ω) $ while at least two positive solutions exist for $λ_{1}(f_{Ω})< λ<λ_{1}(f_Ω)+δ_{a}$ without any assumption on the integral $% \int_{Ω}g(x)ϕ_{1}^{p}dx$, where $λ_{1}(f_{Ω})>0$ is the principal eigenvalue of $-Δ$ in $H_{0}^{1}(Ω)$ with weight function $f_{Ω}:=f|_{Ω}$, and $ϕ_{1}>0$ is the corresponding principal eigenfunction. When $N\geq 3$ and $2<p<\min \{4,2^{\ast }\}$, for $% μ$ sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<λ<λ_{1}(f_{Ω})$; $% (ii)$ under the classical assumption $\int_{Ω}g(x)ϕ_{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $λ_{1}(f_{Ω})\leq λ<λ_{1}(f_Ω)+\overline{δ}% _{a} $; $(iii)$ under the assumption $\int_{Ω}g(x)ϕ_{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $λ^{+}_{a}< λ<λ_{1}(f_Ω)$ for some $a_{0}(p)>0$ and $λ^{+}_{a}\geq0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_08497 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms Sun, Juntao Wang, Kuan-Hsiang Wu, Tsung-fang Analysis of PDEs 35J20, 35J61 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 $N\geq 3,2<p<2^{\ast }:=\frac{2N}{N-2}$, $V\in C(\mathbb{R}^{N})$ is a potential well with the bottom $Ω:=int\{x\in \mathbb{R}^{N}\ |\ V(x)=0\}$. When $N=3$ and $4<p<6$, for each $a>0$ and $μ$ sufficiently large, we obtain that at least one positive solution exists for $% 0<λ\leqλ_{1}(f_Ω) $ while at least two positive solutions exist for $λ_{1}(f_{Ω})< λ<λ_{1}(f_Ω)+δ_{a}$ without any assumption on the integral $% \int_{Ω}g(x)ϕ_{1}^{p}dx$, where $λ_{1}(f_{Ω})>0$ is the principal eigenvalue of $-Δ$ in $H_{0}^{1}(Ω)$ with weight function $f_{Ω}:=f|_{Ω}$, and $ϕ_{1}>0$ is the corresponding principal eigenfunction. When $N\geq 3$ and $2<p<\min \{4,2^{\ast }\}$, for $% μ$ sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<λ<λ_{1}(f_{Ω})$; $% (ii)$ under the classical assumption $\int_{Ω}g(x)ϕ_{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $λ_{1}(f_{Ω})\leq λ<λ_{1}(f_Ω)+\overline{δ}% _{a} $; $(iii)$ under the assumption $\int_{Ω}g(x)ϕ_{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $λ^{+}_{a}< λ<λ_{1}(f_Ω)$ for some $a_{0}(p)>0$ and $λ^{+}_{a}\geq0$. |
| title | On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms |
| topic | Analysis of PDEs 35J20, 35J61 |
| url | https://arxiv.org/abs/2008.08497 |