Abstract
We study the existence of radial solutions for the p-Laplacian Neumann problem with gradient term of the type
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.
Similar content being viewed by others
References
Agarwal, R.P., O’Regan, D., Staněk, S.: Neumann boundary value problems with singularities in a phase variable. Aequ. Math. 69, 293–308 (2005)
Agarwal, R.P., Gala, S., Ragusa, M.A.: A regularity criterion in weak spaces to Boussinesq equations. Mathematics 8, art.n. 920 (2020)
Bonheure, D., Noris, B., Weth, T.: Increasing radial solutions for Neumann problems without growth restrictions. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 573–588 (2012)
Bonheure, D., Grumiau, C., Troestler, C.: Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions. Nonlinear Anal. 147, 236–273 (2016)
Cianciaruso, F., Infante, G., Pietramala, P.: Multiple positive radial solutions for Neumann elliptic systems with gradient dependence. Math. Methods Appl. Sci. 41, 6358–6367 (2018)
Granas, A., Guenther, R.B., Lee, J.W.: Nonlinear Boundary Value Problems for Ordinary Differential Equations, Dissertationes Math. (Rozprawy Mat.) 244, Warsaw (1985)
Kelevedjiev, P.: Existence of solutions for two-point boundary value problems. Nonlinear Anal. 22, 217–224 (1994)
Ma, R., Gao, H., Chen, T.: Radial positive solutions for Neumann problems without growth restrictions. Complex Var. Elliptic Equ. 62, 848–861 (2016)
Ragusa, M.A., Scapellato, A.: Mixed Morrey spaces and their applications to partial differential equations. Nonlinear Anal. 151, 51–65 (2017)
Ragusa, M.A., Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents. Adv. Nonlinear Anal. 9, 710–728 (2020)
Ragusa, M.A.: Commutators of fractional integral operators on Vanishing-Morrey spaces. J. Glob. Optim. 40, 361–368 (2008)
Secchi, S.: Increasing variational solutions for a nonlinear \(p\)-Laplace equation without growth conditions. Ann. Mat. 191, 469–485 (2012)
Serra, E., Tilli, P.: Monotonicity constraints and supercritical Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 63–74 (2011)
Vetro, F.: Infinitely many solutions for mixed Dirichlet–Neumann problems driven by the (\(p, q\))-Laplace operator. Filomat 33, 4603–4611 (2019)
Xing, R., Zhou, B.: Laplacian and signless Laplacian spectral radii of graphs with fixed domination number. Math. Nachr. 288, 476–480 (2015)
Yin, Z.: Monotone positive solutions of second-order nonlinear differential equations. Nonlinear Anal. 54, 391–403 (2003)
Zhang, X., Jiang, J., Wu, Y., Cui, Y.: Existence and asymptotic properties of solutions for a nonlinear Schrödinger elliptic equation from geophysical fluid flows. Appl. Math. Lett. 90, 229–237 (2019)
Acknowledgements
We thank the referees for useful suggestions that helped us to improve the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Maria Alessandra Ragusa.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The project was sponsored by the Education Department of JiLin Province of P. R. China (JJKH20200029KJ).
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-020-01047-x
Keywords
- p-Laplacian Neumann problem
- Existence of radial solutions
- Barrier strip technique
- Topological transversality method