Abstract
We are concerned with the following p-Laplacian equation
where \(-\varDelta _{p}u=div(|\nabla u|^{p-2}\nabla u)\), \(1<p<N\), \(\mu \in {\mathbb {R}}\), \(s\in (\frac{N+2}{N}p,p^{*})\), \(p^{*}=\frac{Np}{N-p}\) is the critical Sobolev exponent. Using constrained variational methods, we prove that the above problem has a normalized solution. Our contribution is that we can deal with the \(L^{2}\)-supercritical case \((\frac{N+2}{N}p ,p^{*})\) by a mountain-pass argument on the prescribed \(L^{2}\)-norm constraint.
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Acknowledgements
The authors are grateful to the referee for their help comments. This work is supported by the National Natural Science Foundation of China (11901514, 11861072, 11961078, 11561072, 11801153) and the Yunnan Province Applied Basic Research for Youths (2018FD085) and the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013) and the Yunnan Province Applied Basic Research for General Project (2019FB001)and the Honghe University Doctoral Research Programs (XJ17B11).
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Communicated by Julian Bonder.
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Wang, W., Li, Q., Zhou, J. et al. Normalized solutions for p-Laplacian equations with a \(L^{2}\)-supercritical growth. Ann. Funct. Anal. 12, 9 (2021). https://doi.org/10.1007/s43034-020-00101-w
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DOI: https://doi.org/10.1007/s43034-020-00101-w