Some aspects of semilinear elliptic boundary value problem (Q2711952)

This paper is devoted to some recent developments in existence theory of the following semilinear elliptic boundary value problem NEWLINE\[NEWLINE\begin{cases} -p\Delta u=f(x,u)\quad & \text{in }\Omega\\ u=0\quad & \text{on } \partial\Omega \end{cases}\tag{1}NEWLINE\]NEWLINE where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with a smooth boundary. The author discusses the superlinear, asymptotically linear and jumping nonlinear problems according to the growth of the function \(g(x,u)\) with respect to \(u\). The author presents a series of new results on the existence of multiple solutions and sign changing solutions for the elliptic boundary value problems.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00021].