Existence results for semilinear elliptic equations with small measure data (Q1611395)

The aim of this paper is to prove, if the data are small enough, existence of a solution for the model problem \[ \begin{cases} -\Delta_p u=f(x)|u |^\gamma+ m\mu\quad &\text{in }\Omega\\ u=0\quad & \text{on }\partial \Omega, \end{cases} \tag{1} \] where \(N\geq 1\), \(\Omega\) is a bounded domain of \(\mathbb R^N\), \(-\Delta_p\) is the \(p\)-Laplacian, \(f\in L^q(\Omega)\), \(q\geq 1\), \(\mu\) is a Radon measure with bounded variation in \(\Omega\) such that \(|\mu |(\Omega)=1\) and \(m\in\mathbb R\). The author presents a smallness condition on \(|m|\) and \(\|f\|_q\), providing existence of a solution of (1).