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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 $\rho ,s\in {L}^{\infty }\left[a,b\right]$ with $\mathrm{ess}{inf}_{\left[a,b\right]}\rho >0,$ $\mathrm{ess}{inf}_{\left[a,b\right]}s>0,$ and consider the boundary value problem

$\left\{\begin{array}{c}-\left(\rho |{x}^{\text{'}}{{|}^{p-2}{x}^{\text{'}}\right)}^{\text{'}}+s\left({|x|}^{p-2}x\right)=\lambda f\left(t,x\right),\hfill \\ \alpha {x}^{\text{'}}\left(a\right)-\beta x\left(a\right)=A,\hfill \\ \gamma {x}^{\text{'}}\left(a\right)-\sigma x\left(a\right)=B,\hfill \end{array}\right\$

where $A,B\in ℝ,$ $\alpha ,\beta ,\gamma ,\sigma >0$, $f:\left[a,b\right]×ℝ\to ℝ$ 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 $I\phantom{\rule{0.166667em}{0ex}}$ 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 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 ${\Phi }-\lambda {\Psi }$, defined on the Sobolev space ${W}^{1,p}\left[a,b\right]$ equipped with the norm

$||x||={\left({\int }_{a}^{b}\left(\rho \left(t\right)|{x}^{\text{'}}{\left(t\right)|}^{p}+{s\left(t\right)|x\left(t\right)|}^{p}\right)dt\right)}^{\frac{1}{p}},$

has at least three critical points for each $\lambda \in I·$

##### MSC:
 34B24 Sturm-Liouville theory 34B15 Nonlinear boundary value problems for ODE 58E30 Variational 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