A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains

By P. J. McKenna, Wolfgang Reichel. 2007

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Abstract A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems − Δ u = f ( x , u ) on a bounded domain Ω ⊂ R n with u = 0 on ∂Ω are studied, where the nonlinearity 0 ⩽ f ( x , s ) grows at most like s p . If Ω is a Lipschitz domain we exhibit two exponents p * and p * , which depend on the boundary behavior of the Green function and on the smallest interior opening angle of ∂Ω. We prove that for 1 p p * all positive very weak solutions are a priori bounded in L ∞ . For p > p * we construct a nonlinearity f ( x , s ) = a ( x ) s p together with a positive very weak solution which does not belong to L ∞ . Finally we exhibit a class of domains for which p * = p * . For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains p * = p * = n + 1 n − 1 is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601–614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49–81].

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/680ea88f16c09394d76f595f66ec526e3c981025

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Citations in the corpus

A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains (2013, methodology): “ |V|m−1V ∈ L1 Φ1 (Ω); • The identity λ Z Ω L−1(V)ψdx= Z Ω |V|m−1Vψdx. (9.2) holds for every test function ψsuch that ψ/Φ1 ∈ L∞(Ω). Remark. The above deﬁnition of very weak solution, has been given in [26], for s= 1. Theorem 9.2 (Bounds and boundary behaviour for the elliptic problem) Let V ≥ 0 be a ”
Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “We expect further progress in this direction with applications to perturbation theory via the so-called 3G theorems, and to nonlinear partial differential equations ([25, 47, 70]).”