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On singular sets of local solutions to \(p\) -Laplace equations. (English) Zbl 1153.35044

From the introduction: We consider local solutions to the following \(p\)-Laplace equation: \[ -\text{div}(|\nabla u|^{p-2}\nabla u)= f,\quad\text{in }\Omega,\tag{1.1} \] where \(2< p<+\infty\), \(\Omega\subset\mathbb{R}^n\) is a bounded domain, and \(f\in L^q(\Omega)\) for some \(q\geq 1\). A local solution of (1.1) means that \(u\in W^{1,p}_{\text{loc}}(\Omega)\) and \[ \int_\Omega|\nabla u|^{p-2}\nabla u\cdot\nabla\varphi\,dx= \int_\Omega f\varphi\,dx,\quad\forall\varphi\in C^\infty_c(\Omega). \] Our main result is the following theorem
Theorem 1.1 Suppose that \(u\in W^{1,p}_{\text{loc}}(\Omega)\) is a local solution to (1.1), \(f\in L^q(\Omega)\), and \[ q> {n\over p},\quad q\geq 2. \] Then \(f(x)= 0\), a.e. \(x\in\{\nabla u= 0\}\).

MSC:

35J70 Degenerate elliptic equations
49J45 Methods involving semicontinuity and convergence; relaxation
35B37 PDE in connection with control problems (MSC2000)
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