Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities
For the Dirichlet boundary value problem (1) \(-\Delta u= p(u)\) conditions on \(p\) related to the eigenvalues of \(-\Delta u= \lambda u\) are found, guaranteeing existence of multiple solutions. For the problem at resonance, i.e., if \(\lim_{|t|\to \infty}p(t)/t= \lambda\) exists and equals to an eigenvalue, one has \(p(u)= \lambda u+ g(u)\), where \(\lim_{|t|\to \infty} g(t)/t= 0\). The growth condition \(\lim_{|t|\to \infty} tg(t)= -\infty\) and certain additional constraints on the global behaviour of \(g(t)/t\) are shown to imply existence of at least two (non-trivial) solutions to (1). An analogous results is proved when \(\lim_{|t|\to \infty} tg(t)= \infty\). A multiple existence result is also proved for \(g\) bounded with limits at \(+ \infty\) and \(-\infty\), satisfying the (milder) Landesman-Lazer conditions. Further, conditions guaranteeing infinitely many solutions are presented. As to the jumping nonlinearities, that is, \(-\Delta u= au^+ -bu^- + g(u)\), where \(a= \lim_{t\to \infty} p(t)/t\), \(b= \lim_{t\to -\infty}p(t)/t\), \(a< \lambda< b\) for some eigenvalue \(\lambda\), a multiple existence result is proved for a bounded \(g\) and \((a, b)\) outside the extended spectrum of \(-\Delta\). In particular, the interval \((a, b)\) can contain a finite set of eigenvalues of \(-\Delta\). The above described results are based on the author's abstract multiple existence theorem.
scientific article; zbMATH DE number 566492