Global bifurcation results for semilinear elliptic equations on \(\mathbb{R}^N\): The Fredholm case
The nonlinear eigenvalue problem \[ -\Delta u=\lambda g(x)f(u)\quad \text{in } \mathbb{R}^N\;(N\geq 3),\;0<u<1,\;\lim_{| x|\to 0} u(x)=0 \] is studied. Here \(g\) is an \(L^\infty\) smooth function such that \(g(x)>0\) on \(\Omega^+\) with \(| \Omega^+ |>0\) and there exist \(k\), \(R_0>0\) such that \(g(x) <-k\) if \(| x| >R_0\), and \(f:[0,1] \to\mathbb{R}^+\) is a smooth function such that \(f(0)=f(1) =0\), \(f'(0) >0\) \(f'(1) <0\) and \(f(u) >0\) if \(u\in (0,1)\), satisfying additional regularity conditions. The main result states that under these assumptions, and if \(N= 3,4,5\), a global bifurcation theorem of Rabinowitz's type holds, giving the existence of a branch of positive solutions such that \(0<u<1\). In Section 2 the problem is settled in the framework of Sobolev spaces and an extension of Poincaré's inequality is proved. The associated eigenvalue problem is considered in Section 3 and the existence of a principal eigenvalue with a positive eigenfunction in \(H^2\) is shown. The existence of a local branch of positive solutions is proved in Section 4 by using Crandall-Rabinowitz's theorem. The difficulty raised by the noncompactness of the Fredholm operators involved is overcomed in Section 5 by using an extension of degree theory due to Dancer and an important result concerning the compactness of the solution set is given in Section 6. Finally, Section 7 is devoted to the main bifurcation theorem and some complementary results.
scientific article; zbMATH DE number 1130545
scientific article