On positive solutions of a nonlinear equilibrium boundary value problem
The author obtains a necessary and sufficient condition for the existence of positive solutions to the boundary value problem \[ \Delta u+uf(u)- uv\phi (u)=0,\quad d\Delta v+g(v)+kuv\phi (u)=0\quad in\quad \Omega \subset R^ n,\quad u=1,\quad v=0\quad on\quad \partial \Omega, \] where i) \(f(u)>0\) for \(0\leq u<c_ 0\), \(f(c_ 0)=0\), \(f(u)<0\) for \(c_ 0<u,\) ii) \(u\phi (u)>0\), \([u\phi (u)]'>0\), and iii) \(g(v)<g'(0)v\), \(g(0)=0\), \(g'(0)<0\), \(| g(v)/v| \to \infty\) as \(v\to \infty.\) The condition is that \(g'(0)+k\phi (1)>d\lambda_ 1,\) where \(\lambda_ 1\) is the first fixed membrane eigenvalue. The proof of the result is accomplished by degree theory and the modification of a theorem due to \textit{E. N. Dancer} [J. Math. Anal. Appl. 91, 131-151 (1983; Zbl 0512.47045)]. This work complements earlier work of the author [Trans. Am. Math. Soc. 305, No.1, 143-166 (1988; Zbl 0655.35021)] and \textit{P. DeMottoni} and \textit{F. Rothe} [SIAM J. Appl. Math. 37, 648-663 (1979; Zbl 0425.35055)].
scientific article; zbMATH DE number 4117041
scientific article