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On Stable Solutions to Weighted Quasilinear Problems of Gelfand Type

Abstract

Let \(p \ge 2\) and \(w,f \in L^1_{loc}(\mathbb {R}^N)\) be nonnegative functions such that \(w(x) \le C_1|x|^a\) and \(f(x) \ge C_2|x|^b\) for large |x|. We prove the Liouville type theorem for stable \(W^{1,p}_{loc}\) solutions of weighted quasilinear problem

$$\begin{aligned} -\text {div} (w(x)|\nabla u|^{p-2} \nabla u) = f(x) e^u \quad \text {in } \mathbb {R}^N. \end{aligned}$$

The result holds true for \(N < \dfrac{(p-a)(p+3) + 4b}{p-1}\) and is sharp in the case that w and f are Hardy–Hénon potentials. We also prove the full classification of solutions which are stable outside a compact set to Gelfand equation \(-\Delta _N u = e^u\) in \(\mathbb {R}^N\).

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Correspondence to Phuong Le.

Additional information

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.02-2017.307.

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Le, P., Le, D.H.T. & Le, K.A.T. On Stable Solutions to Weighted Quasilinear Problems of Gelfand Type. Mediterr. J. Math. 15, 94 (2018). https://doi.org/10.1007/s00009-018-1143-7

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Keywords

  • Quasilinear problems
  • stable solutions
  • Liouville theorems
  • Gelfand nonlinearity

Mathematics Subject Classification

  • 35B53
  • 35J92
  • 35B08
  • 35B35