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Radial Solutions for p-Laplacian Neumann Problems Involving Gradient Term Without Growth Restrictions

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Abstract

We study the existence of radial solutions for the p-Laplacian Neumann problem with gradient term of the type

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _{p}u=f(|x|,u,x\cdot \nabla u)\quad \text {in} ~\varOmega ,\\ \displaystyle \frac{\partial u}{\partial \mathbf{n} }=0\quad \text {on}~ \partial \varOmega , \end{array} \right. \end{aligned}$$

where \(\Delta _pu=\text {div}(|\nabla u|^{p-2}\nabla u)\) is the p-Laplace operator with \(p>1\), \(\varOmega \subset \mathbb {R}^N(N\ge 2)\) is a ball. We do not impose any growth restrictions on the nonlinearity. By using the topological transversality method together with the barrier strip technique, the existence of radial solutions to the above problem is obtained.

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Acknowledgements

We thank the referees for useful suggestions that helped us to improve the presentation of the paper.

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Correspondence to Libo Wang.

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Communicated by Maria Alessandra Ragusa.

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The project was sponsored by the Education Department of JiLin Province of P. R. China (JJKH20200029KJ).

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Pei, M., Wang, L. & Lv, X. Radial Solutions for p-Laplacian Neumann Problems Involving Gradient Term Without Growth Restrictions. Bull. Malays. Math. Sci. Soc. 44, 2035–2047 (2021). https://doi.org/10.1007/s40840-020-01047-x

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  • DOI: https://doi.org/10.1007/s40840-020-01047-x

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