The aim of this paper is to study the existence of ground states for a class of fractional Kirchhoff type equations with critical or supercritical nonlinearity
$ (a+b\int_{\mathbb{R}^{3}}|(-\bigtriangleup)^{\frac{s}{2}}u|^{2}dx)(-\bigtriangleup)^{s}u = \lambda u +|u|^{q-2 }u+\mu|u|^{p-2}u, \ x\in\mathbb{R}^{3}, $
with prescribed $ L^{2} $-norm mass
$ \int_{\mathbb{R}^{3}}u^{2}dx = c^{2} $
where $ s\in(\frac{3}{4}, \ 1), \ a, b, c > 0, \ \frac{6+8s}{3} < q < 2_{s}^{\ast}, \ p\geq 2^{\ast}_{s}\ (2^{\ast}_{s} = \frac{6}{3-2s}), \ \mu > 0 $ and $ \lambda\in \mathbb{R} $ as a Langrange multiplier. By combining an appropriate truncation argument with Moser iteration method, we prove that the existence of normalized solutions for the above equation when the parameter $ \mu $ is sufficiently small.
Citation: Huanhuan Wang, Kexin Ouyang, Huiqin Lu. Normalized ground states for fractional Kirchhoff equations with critical or supercritical nonlinearity[J]. AIMS Mathematics, 2022, 7(6): 10790-10806. doi: 10.3934/math.2022603
The aim of this paper is to study the existence of ground states for a class of fractional Kirchhoff type equations with critical or supercritical nonlinearity
$ (a+b\int_{\mathbb{R}^{3}}|(-\bigtriangleup)^{\frac{s}{2}}u|^{2}dx)(-\bigtriangleup)^{s}u = \lambda u +|u|^{q-2 }u+\mu|u|^{p-2}u, \ x\in\mathbb{R}^{3}, $
with prescribed $ L^{2} $-norm mass
$ \int_{\mathbb{R}^{3}}u^{2}dx = c^{2} $
where $ s\in(\frac{3}{4}, \ 1), \ a, b, c > 0, \ \frac{6+8s}{3} < q < 2_{s}^{\ast}, \ p\geq 2^{\ast}_{s}\ (2^{\ast}_{s} = \frac{6}{3-2s}), \ \mu > 0 $ and $ \lambda\in \mathbb{R} $ as a Langrange multiplier. By combining an appropriate truncation argument with Moser iteration method, we prove that the existence of normalized solutions for the above equation when the parameter $ \mu $ is sufficiently small.
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Huanhuan Wang, Kexin Ouyang, Huiqin Lu. Normalized ground states for fractional Kirchhoff equations with critical or supercritical nonlinearity[J]. AIMS Mathematics, 2022, 7(6): 10790-10806. doi: 10.3934/math.2022603
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