Exact multiplicity and stability of solutions of a 1-dimensional, p-Laplacian problem with positive convex nonlinearity. (June 2019) - Rynne, Bryan P.
Abstract: We consider the p -Laplacian problem (1) − ϕ p ( u ′ ) ′ = λ f ( u ), on ( − 1, 1 ), (2) u ( ± 1 ) = 0, where p > 1 ( p ≠ 2 ), ϕ p ( z ) ≔ | z | p − 1 sgn z, z ∈ R, λ ⩾ 0, and f : R → ( 0, ∞ ) is C 2 and there exists C > 0, q > p such that f ( ξ ) ⩾ C ( 1 + ξ q − 1 ), f ′ ( ξ ) > 0, f ′ ′ ( ξ ) > 0, ξ > 0 . We show that the set of solutions ( λ, u ) of(1) –(2) consists of a C 2 curve in ( 0, ∞ ) × C 1 [ − 1, 1 ] with end point ( 0, 0 ) ∈ R × C 1 [ − 1, 1 ] and which tends to 0 × ∞, and has a single turning point. Thus, there exists λ ∗ > 0 such that(1) –(2) has exactly two solutions, u − ( λ ) < u + ( λ ), when 0 < λ < λ ∗, and no solutions when λ > λ ∗ . When 0 < λ < λ ∗ the solutions u ± ( λ ) are equilibria of a related time-dependent, parabolic problem, and in this context their stability is of interest. We show that the 'lower' solution u − ( λ ) is stable and the 'upper' solution u + ( λ ) is unstable, and solutions of the parabolic problem with initial values below u + ( λ ) converge to u − ( λ ), while those with initial values above u + ( λ ) are unbounded.