Strong stability results for solutions of elliptic equations with power-like lower order terms and measure data
This paper deals with extending a well-known result by H. Brezis to the nonlinear case, namely: Let \(u_n\) be the sequence of solutions of \[ \begin{cases} -\text{div} \bigl(a(x,u_n, \nabla u_n) \bigr)+ |u_n|^{q-1} \cdot u_n=f_n \quad &\text{in }\Omega\\ u_n=0 \quad &\text{on }\partial \Omega, \end{cases} \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), and \(f_n\) is a sequence of functions which is strongly convergent to a function \(f\) in \(L^1_{\text{loc}}(\Omega \setminus K)\), with \(K\) compact in \(\Omega\) of zero \(r\)-capacity; no assumptions are made on \(f_n\) on the set \(K\). Under some suitable additional assumptions on the data of (1), the authors show that \(u_n\) converges to \(u\), which verifies the same problem with datum \(f\).
elliptic equation with power-line lower order term
scientific article; zbMATH DE number 1797581