Existence and Qualitative Properties of Radial Solutions to a Nonlinear Equation on Lifshitz Black Hole Backgrounds

Document Type : Research Article

Author

Department of Physics, K.L.S. College, Nawada, Magadh University, Bodh Gaya, Bihar 805110, India

Abstract

We study a nonlinear scalar equation on a $(d{+}2)$--dimensional Lifshitz black hole background with dynamical critical exponent $z>1$.
For static, radially symmetric configurations the Klein--Gordon equation with power-type self-interaction reduces to a non-autonomous second-order ordinary differential equation on the half-line. We formulate the problem in divergence form, identify a natural weighted energy, and prove existence of bounded solutions decaying at infinity via a compactness argument and a monotone iteration scheme built upon the positive Green operator of the linearized problem. We also derive sharp near-horizon and asymptotic expansions, and we discuss uniqueness under a smallness condition on the nonlinearity. An explicit family of backgrounds $f(r)=1-(r_h/r)^{d+z}$ is used to illustrate the boundary behavior and to present a numerically robust shooting strategy consistent with the analysis.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 10, Issue 2
December 2025
Pages 200-218

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History

  • Receive Date: 12 October 2025
  • Accept Date: 15 December 2025
  • Publish Date: 30 December 2025

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