On the Poisson equation of p-Laplacian and the nonlinear Hardy-type problems

Abstract

In this note, we show that in some cases, via the use of Hardy-type inequality, there is a non-trivial nonnegative \(W^{1,p}(R^n)\) weak solution to quasi-linear elliptic problem with the p-Laplacian on \(R^n\) and with Hardy-type singularity term. We also study the behavior of solutions to the Poisson equation of p-Laplacian on the whole space and this Poisson equation has a close relationship with the Gelfand-type equation.

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Correspondence to Li Ma.

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The research is partially supported by the National Natural Science Foundation of China (No. 11771124). We certify that the general content of the manuscript, in whole or in part, is not submitted, accepted, or published elsewhere, including conference proceedings.

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Ma, L. On the Poisson equation of p-Laplacian and the nonlinear Hardy-type problems. Z. Angew. Math. Phys. 72, 34 (2021). https://doi.org/10.1007/s00033-020-01465-8

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Keywords

  • p-Laplacian
  • Ground state
  • Existence of positive solutions
  • Rearrangement
  • Variational method
  • Nehari manifold

Mathematics Subject Classification

  • 35A05
  • 35A15
  • 35B50
  • 35J60
  • 46Txx
  • 53C70
  • 58E50