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Multiplicity results for Sturm-Liouville boundary value problems. (English) Zbl 1175.34028

The authors establish a result on existence of multiple solutions for a particular class of second order Sturm-Liouville boundary value problems. To be more precise, let p>1, let ρ,sL [a,b] with ess inf [a,b] ρ>0, ess inf [a,b] s>0, and consider the boundary value problem

-(ρ|x ' | p-2 x ' ) ' +s|x| p-2 x=λf(t,x),αx ' (a)-βx(a)=A,γx ' (a)-σx(a)=B,

where A,B, α,β,γ,σ>0, f:[a,b]× is an L 1 -Carathérodory function, and λ is a positive real parameter. Then the main result of the paper (Theorem 3.1) provides sufficient conditions on f, p, s in order to ensure the existence of an open interval I for which the above problem has at least three weak solutions whenever λI·

It is worth pointing out that Theorem 3.1 improves a result of Y. Tian and W. Ge [Rocky Mountain J. Math. 38, 309–327 (2008; Zbl 1171.34019)] in the sense that its assumptions are much simpler than those of the above mentioned paper.

The proof of Theorem 3.1 is based on the fact that an adequate (coercive) functional Φ-λΨ, defined on the Sobolev space W 1,p [a,b] equipped with the norm

||x||= a b (ρ(t)|x ' (t)| p +s(t)|x(t)| p )dt 1 p ,

has at least three critical points for each λI·

MSC:
34B24Sturm-Liouville theory
34B15Nonlinear boundary value problems for ODE
58E30Variational principles on infinite-dimensional spaces
References:
[1]Bonanno, G.; Candito, P.: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. diff. Eq. 244, 3031-3059 (2008) · Zbl 1149.49007 · doi:10.1016/j.jde.2008.02.025
[2]G. Bonanno, S.A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, preprint. · Zbl 1194.58008 · doi:10.1080/00036810903397438
[3]Du, Z.; Lin, X.; Tisdell, C.: A multiplicity result for p-Laplacian boundary value problems via critical points theorem, Appl. math. Comput. 205, 231-237 (2008) · Zbl 1173.34007 · doi:10.1016/j.amc.2008.07.011
[4]Tian, Y.; Ge, W.: Second-order Sturm – Liouville boundary value problem involving the one-dimensional p-Laplacian, Rocky mountain J. Math. 38, 309-327 (2008) · Zbl 1171.34019 · doi:10.1216/RMJ-2008-38-1-309