Structure of boundary blow-up solutions for quasi-linear elliptic problems. II: Small and intermediate solutions
Let \(\Omega\subset \mathbb R^N\) be a bounded smooth domain. The authors deal with a boundary blow-up solutions of the problem \[ \begin{gathered} -\Delta_p u=\lambda f(u)\quad\text{in }\Omega,\\ u|_{\partial\Omega}= \infty,\end{gathered}\tag{1} \] where \(\lambda> 0\) is a parameter, \(\Delta_pu= \text{div}(|Du|^{p-2} Du)\) with \(1<p<\infty\) is the \(p\)-Laplacian. Under natural assumptions on \(f(s)\) the authors discuss the structure of small and intermediate solutions of (1). They show that, for sufficiently large \(\lambda\), the equation (1) has a unique small solution and infinitely many intermediate solutions. It is important to emphasize that, in contrast to previous results, a strong monotone assumption on the nonlinearity was not imposed. Part I, see Proc. R. Soc. Edinb., Sect. A, Math. 135, No. 3, 615--642 (2005; Zbl 1129.35381).
scientific article; zbMATH DE number 2157588
scientific article