Some bifurcation results for rapidly growing nonlinearities (Q1943264)

Let \(\Omega\) be a bounded smooth domain in \({\mathbb R}^N\). This paper deals with the quasilinear elliptic problem \[ -\Delta u=\lambda f(u)\qquad\text{in}\;\Omega, \] under the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\). In the first part it is assumed that \(f(t)\) behaves like \(\exp t^\alpha\) as \(t\rightarrow\infty\), for some \(\alpha>0\). The main result in this case establishes that the upper limit of the dimension \(N\) is independent of \(\alpha\). This is a surprising difference from variational methods where the size of \(\alpha\) often seems important. Next, it is studied the case where \(f(t)=(1-\varepsilon t)^k\exp t\), with \(\varepsilon>0\). Such nonlinearities frequently occur in both the theory of catalysts and combustion theory. The author proves that there is a connected branch of positive solutions with many bifurcations. The method developed in the present paper works for many other small perturbations of \(\exp t\), even many nonlocal ones. In the final part it is established the existence of an unbounded closed connected subset of the set of solutions which has infinitely many solutions and these are unbounded.