Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains

AbstractWe consider nonlinear elliptic eigenvalue problems on unbounded domains G⊆Rn. Using an extended Ljusternik-Schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in L2(G). Moreover, we establish that the infimum λ∗ of the spectrum of the linearized problem L is always a bifurcation point. In addition, there is an infinity of branches emanating at λ∗ from the trivial line of solutions if λ∗ belongs to the essential spectrum of L