\(L^ \infty\) boundedness of nonlinear eigenfunction under singular variation of domains
The weak form of the nonlinear eigenvalue problem \[ - \Delta v_\varepsilon(x)= \lambda(\varepsilon) v_\varepsilon(x)^p,\quad p\in (0, 2), \] is studied in a domain \(M_\varepsilon= M- \overline B_\varepsilon\), where \(M\) has a smooth boundary and \(B_\varepsilon\) is a ball of radius \(\varepsilon\). The paper shows that the positive eigenfunctions are \(L^\infty\)-bounded by a constant which does not depend on \(\varepsilon\).
scientific article; zbMATH DE number 880176