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Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces

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Abstract

We prove a weak comparison principle in narrow unbounded domains for solutions to \(-\Delta _p u=f(u)\) in the case \(2<p< 3\) and \(f(\cdot )\) is a power-type nonlinearity, or in the case \(p>2\) and \(f(\cdot )\) is super-linear. We exploit it to prove the monotonicity of positive solutions to \(-\Delta _p u=f(u)\) in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.

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Notes

  1. Note that here \(B_r({x^{\prime }})\) is the ball in \(\mathbb R ^{N-1} \)of radius \(r\) centered at \({x^{\prime }}\).

  2. \(d_0\) will actually depend on the Lipschitz constant \(L_f\) of \(f\) in the interval \([-\max \{\Vert u\Vert _\infty ,\Vert v\Vert _\infty \} ,\max \{\Vert u\Vert _\infty ,\Vert v\Vert _\infty \}]\).

  3. The case \(N=2\) has been already considered in [11].

  4. In the case \(f(0)>0\), the solution does not exist at all.

  5. Note that the condition \(\gamma > N\,-\,2t\) holds true for \(r\approx 1\) and \(\gamma \approx N-2\) that we may assume with no loose of generality.

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Acknowledgments

The authors would like to thank the anonymous referees for their useful comments and suggestions.

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Authors and Affiliations

  1. LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Amiens, France

    Alberto Farina

  2. Dipartimento di Matematica, Università della Calabria, Ponte Pietro Bucci 31B, Arcavacata di Rende, 87036 , Cosenza, Italy

    Luigi Montoro & Berardino Sciunzi

Authors
  1. Alberto Farina
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  2. Luigi Montoro
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  3. Berardino Sciunzi
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Corresponding author

Correspondence to Berardino Sciunzi.

Additional information

In memory of James B. Serrin.

AF and BS were partially supported by ERC-2011-grant: Elliptic PDE’s and symmetry of interfaces and layers for odd nonlinearities. LM and BS were partially supported by PRIN-2011: Variational and Topological Methods in the Study of Nonlinear Phenomena. LM has been partially supported by project MTM2010-18128 of MICINN Spain.

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Farina, A., Montoro, L. & Sciunzi, B. Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces. Math. Ann. 357, 855–893 (2013). https://doi.org/10.1007/s00208-013-0919-0

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  • DOI: https://doi.org/10.1007/s00208-013-0919-0

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