Positive solutions of nonlinear elliptic problems approximating degenerate equations (Q1922774)

This article concerns the effect of geometry on the multiplicity of positive solutions to semilinear elliptic problems of type \[ -\text{div}(a_\varepsilon(x)\nabla u)= g(x,u)\quad\text{in }\Omega,\quad u|_{\partial\Omega}= 0\leqno(1)_\varepsilon \] in a smooth bounded domain \(\Omega\subset\mathbb{R}^N\), \(N\geq 3\), where \(\varepsilon\) is a positive parameter, \(a_\varepsilon\) is a positive definite symmetric \(N\)-square matrix function a.e. in \(\Omega\) with \(L^\infty(\Omega,\mathbb{R})\) entries, and \(g(x,t)\) is a real-valued function in \(\Omega\times\mathbb{R}\) which is measurable with respect to \(x\), differentiable, superlinear, and subcritical with respect to \(t\). The number of positive solutions to \((1)_\varepsilon\) depends on the number of disjoint ``degeneration subsets'' for \(a_\varepsilon\), defined to be smooth subdomains \(\Omega_i\) with \(\overline\Omega_i\subseteq\overline\Omega^*_i\subset \Omega\) for every \(i= 1,\dots,k\geq 2\) and (typically) simply-connected pairwise disjoint subsets \(\overline\Omega^*_i\) such that \(a_\varepsilon(x)\to 0\) as \(\varepsilon\to 0+\) uniformly in \(\bigcup^k_{i=1}\Omega_i\). For this structure, the main theorem states that \((1)_\varepsilon\) has at least \(k+1\) distinct positive solutions for all sufficiently small \(\varepsilon>0\). In particular, this result applies to the case \(g(x,t)= t^{p-1}\), \(2<p<2N/(N- 2)\). The procedure involves critical point theory of the energy functional associated with \((1)_\varepsilon\) on the unit ball in \(L^p(\Omega)\). Qualitative results on the behavior of the solutions as \(\varepsilon\to 0+\) are included. A bibliography of 28 items emphasizes multiplicity theory during 1984-1994.