Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points. (English) Zbl 0851.35047

The author studies quasilinear elliptic equations of type \(- \text{div } A(x, u, \nabla u)+ B(x, u, \nabla u)= \mu\), where \(A\) and \(B\) are Carathéodory functions with growth exponent \(p\in (1, \infty)\) and \(\mu\in (W^{1, p}_0(\Omega))^*\) is a nonnegative Radon measure. Let \(u\) be a weak solution of such equation; the author estimates \(u(x)\) at an interior point \(x\in \Omega\) or an irregular point \(x\in \partial \Omega\), in terms of a norm of \(u\), a nonlinear potential of \(\mu\) and the Wiener integral of \(\mathbb{R}^n\backslash \Omega\). This result quantifies the result on necessity of the Wiener criterion.


35J70 Degenerate elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
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