Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity
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| Format: | Preprint |
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2022
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| _version_ | 1866912076946997248 |
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| author | Zhang, Jian Zhang, Jianjun Zhong, Xuexiu |
| author_facet | Zhang, Jian Zhang, Jianjun Zhong, Xuexiu |
| contents | In this paper, we are concerned with normalized solutions of the Kirchhoff type equation
\begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2\mathrm{d} x\right)Δu = λu +f(u) \ \ \mathrm{in} \ \ \mathbb{R}^N \end{equation*} with $u \in S_c:=\left\{u \in H^1(\R^N): \int_{\R^N}u^2 \mathrm{d}x=c^2\right\}$. When $N=2$ and $f$ has exponential critical growth at infinity, normalized mountain pass type solutions are obtained via the variational methods. When $N \ge 4$, $M(t)=a+bt$ with $a$, $b>0$ and $f$ has Sobolev critical growth at infinity, we investigate the existence of normalized ground state solutions and normalized mountain pass type solutions. Moreover, the non-existence of normalized solutions is also considered. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_12911 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity Zhang, Jian Zhang, Jianjun Zhong, Xuexiu Analysis of PDEs In this paper, we are concerned with normalized solutions of the Kirchhoff type equation \begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2\mathrm{d} x\right)Δu = λu +f(u) \ \ \mathrm{in} \ \ \mathbb{R}^N \end{equation*} with $u \in S_c:=\left\{u \in H^1(\R^N): \int_{\R^N}u^2 \mathrm{d}x=c^2\right\}$. When $N=2$ and $f$ has exponential critical growth at infinity, normalized mountain pass type solutions are obtained via the variational methods. When $N \ge 4$, $M(t)=a+bt$ with $a$, $b>0$ and $f$ has Sobolev critical growth at infinity, we investigate the existence of normalized ground state solutions and normalized mountain pass type solutions. Moreover, the non-existence of normalized solutions is also considered. |
| title | Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2210.12911 |