A critically degenerate elliptic Dirichlet problem, spectral theory and bifurcation. (January 2020) - Stuart, C.A.
Abstract: In an open, bounded subset Ω of R N such that 0 ∈ Ω we consider the nonlinear eigenvalue problem − ∑ i, j = 1 N ∂ i { A i j ( x ) ∂ j u } + V ( x ) u + n ( x, ∇ u ) + g ( x, u ) = λ u in Ω ∫ Ω u 2 + ∑ i, j = 1 N A i j ∂ j u ∂ i u d x < ∞ and u = 0 on ∂ Ω, where V ∈ L ∞ ( Ω ) and the nonlinear terms n and g are of higher order near 0 so that the formal linearization about the trivial solution u ≡ 0 is − ∑ i, j = 1 N ∂ i { A i j ( x ) ∂ j u } + V u = λ u . The leading term is degenerate elliptic on Ω because it is assumed that there are constants C 2 ≥ C 1 > 0 such that C 1 | x | 2 | ξ | 2 ≤ ∑ i, j = 1 N A i j ( x ) ξ j ξ i ≤ C 2 | x | 2 | ξ | 2 for all ξ ∈ R N and almost all x ∈ Ω . This is the lowest level of degeneracy at x = 0 for which the linearization has a non-empty essential spectrum. Furthermore, elliptic regularity theory does not apply at x = 0 . Eigenfunctions of the linearization and solutions of the nonlinear problem having finite energy may be singular at the origin. The main results establish conditions for the existence or not of eigenvalues of the linearization, describe the behaviour of eigenfunctions as x → 0 and determine values of the parameter λ at which bifurcation from the line of trivial solutions occurs. Standard bifurcation theory does not apply, even when n and g are smooth functions, since the nonlinear terms generate operators which are Gâteaux but not Fréchet differentiable at the trivial solution.