Global bifurcation for asymptotically linear Schrödinger equations

摘要

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schrödinger equations ({left{begin{array}{lll}Delta u + f(x, u)u = lambda u quad {rm in} ; mathbb{R}^N, u in H^1(mathbb{R}^N) backslash {0}, quad N ; geqslant ; 1.qquadqquadqquad(1)end{array}right.}) The method is topological, based on recent developments of degree theory. We use the inversion ({uto v:= u/Vert uVert_X^2}) in an appropriate Sobolev space ({X=W^{2,p}(mathbb{R}^{N}),}) and we first obtain bifurcation from the line of trivial solutions for an auxiliary problem in the variables ({(lambda,v) in {mathbb R}times X.}) This problem has a lack of compactness and of regularity, requiring a truncation procedure. Going back to the original problem, we obtain global branches of positiveegative solutions ‘bifurcating from infinity’. We believe that, for the values of λ covered by our bifurcation approach, the existence result we obtain for positive solutions of (1) is the most general so far.