Abstract
We examine the regularity of the extremal solution of the nonlinear eigenvalue problemon a general bounded domainΩin ℝN, with Navier boundary conditionu= Δuon ∂Ω. Firstly, we prove the extremal solution is smooth for anyp> 1 andN⩽ 4, which improves the result of Guo and Wei (Discrete Contin. Dynam. Syst.A34(2014), 2561–2580). Secondly, ifp= 3,N= 3, we prove that any radial weak solution of this nonlinear eigenvalue problem is smooth in the caseΩ= 𝔹, which completes the result of Dávilaet al. (Math. Annalen348(2009), 143–193). Finally, we also consider the stability of the entire solution of Δ2u= –l/upin ℝNwithu> 0.
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