HIGHLIGHTS
SUMMARY
The eigenfunctions are dense in H10 (Ω). (ii) If κ ∈ L∞ (Ω) then each of the eigenvalues λ j is a bifurcation point for Eq with respect to the trivial solution ¢ {(0, λ): λ ∈ R}. By definition, this means that for all ¡ family j ∈ Z there is a sequence u nj, λnj n∈N ⊂ H10 (Ω) {0} × R of nontrivial solutions such that (u nj, λnj ) → (0, λ j ) in H10 (Ω)×R as n → ∞. In a one-dimensional case, for any given λ ∈ R there are infinitely many nontrivial solutions for Eq provided that κ ∈ L∞ (Ω) is uniformly positive or uniformly negative. For j . . .
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