L^∞ a-priori estimates for subcritical p-laplacian equations with a Carathéodory nonlinearity

2022/09/14 by Pardo, Rosa
#35A23 #35B45 #35J25 #35J92 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.06568

Abstract

We present new L^∞ a priori estimates for weak solutions of a wide class of subcritical p-laplacian equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in elliptic regularity for the p-laplacian combined either with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a quasilinear boundary value problem -Δ_p u= f(x,u), in Ω, with Dirichlet boundary conditions, where Ω⊂ ℝ^N , with p0 there exists a constant C_ε>0 such that for any solution u∈ H¹₀(Ω), the following holds [log(e+‖u‖_∞)]^α≤ C_ε (1+‖u‖_p^*) ^{(p^*_μ-p)(1+ε)} , where C_ε is independent of the solution u.

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