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Necessary conditions of existence for an elliptic equation with source term and measure data involving \(p\)-Laplacian. (English) Zbl 1114.35316

Summary: We study the nonnegative solutions to equation \[ -\Delta_p u=u^q+ \lambda\nu, \] in a bounded domain \(\Omega\) of \(\mathbb R^N\), where \(1< p< N\), \(q> p- 1\), \(\nu\) is a nonnegative Radon measure on \(\Omega\), and \(\lambda>0\) is a parameter. We give necessary conditions on \(\nu\) for existence, with \(\lambda\) small enough, in terms of capacity. We also give a priori estimates of the solutions.

MSC:

35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
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