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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 $\rho ,s\in L^\infty [a,b]$ with $\mathrm{ess}\inf_{[a,b]}\rho >0,$ $\mathrm{ess} \inf_{[a,b]}s>0,$ and consider the boundary value problem $$\cases -(\rho |x^{\prime }|^{p-2}x^{\prime })^{\prime }+s\left( |x|^{p-2}x\right) =\lambda f(t,x), \\ \alpha x^{\prime }(a)-\beta x(a)=A, \\ \gamma x^{\prime }(a)-\sigma x(a)=B, \endcases $$ where $A,B\in \Bbb{R},$ $\alpha ,\beta ,\gamma ,\sigma >0$, $f:[a,b]\times \Bbb{R}\rightarrow \Bbb{R}$ is an $L^1$-Carathérodory function, and $\lambda $ 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 $\lambda\in I.$ It is worth pointing out that Theorem 3.1 improves a result of {\it Y. Tian} and {\it 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 $\Phi -\lambda \Psi $, defined on the Sobolev space $W^{1,p}[a,b]$ equipped with the norm $$||x||=\left( \int_a^b(\rho (t)|x^{\prime }(t)|^p+s(t)|x(t)|^p)dt\right) ^{\frac 1p},$$ has at least three critical points for each $\lambda \in I.$

34B24Sturm-Liouville theory
34B15Nonlinear boundary value problems for ODE
58E30Variational principles on infinite-dimensional spaces
Full Text: DOI
[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