A priori Estimates and Existence for Elliptic Systems via Bootstrap in Weighted Lebesgue Spaces

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A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems (2007, methodology): “ve been established for subcritical, superlinear nonlinearities via diﬀerent methods by Brezis, Turner [7], Gidas, Spruck [13], DeFigueiredo, Lions and Nussbaum [10] and recently by Quittner, Souplet [20] and McKenna, Reichel [19]. In the higher-order case m≥ 2 the theory is far less developed and s”
L∞-Estimates for nonlinear elliptic Neumann boundary value problems (2010, result): “with p;q > 0 and the same assumptions on and . To obtain a priori estimates for elliptic systems of Laplace type we also refer to the publications in [11] and [ 28 ].”
The proof of the Lane–Emden conjecture in four space dimensions (2009, unknown): “Quittner and Souplet [18] and see also Souplet [24, Remarks 3.1(b) and 3.2]) that condition (1.5) corresponds to a critical regularity exponent, related to the possibility of reaching L ∞ from an L 1 a priori estimate on the right-hand side of (1.3).”