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Zbl 1175.34028
Bonanno, Gabriele; Riccobono, Giuseppa
Multiplicity results for Sturm-Liouville boundary value problems.
(English)
[J] Appl. Math. Comput. 210, No. 2, 294-297 (2009). ISSN 0096-3003

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 $\lambda\in I.$ \par 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. \par 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 $\lambda \in I.$
[Antonio Linero Bas (Murcia)]
MSC 2000:
*34B24 Sturm-Liouville theory
34B15 Nonlinear boundary value problems of ODE
58E30 Variational principles on infinite-dimensional spaces

Keywords: Boundary value problem; Sturm-Liouville problem; weak solutions; classical solutions; multiple solutions; functional; critical points; Sobolev spaces; variational methods

Citations: Zbl 1171.34019

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