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Multiplicity results for classes of one-dimensional $$p$$-Laplacian boundary-value problems with cubic-like nonlinearities. (English) Zbl 0952.34007
Summary: The author studies boundary value problems of the type $-(\varphi_{p}( u'))'=\lambda f(u),\text{ in }(0,1),\quad u(0)=u( 1)=0,$ with $$p>1$$, $$\varphi_{p}(x) =\left|x\right|^{p-2}x$$, and $$\lambda >0$$. He provides multiplicity results when $$f$$ behaves like a cubic with three distinct roots, at which it satisfies Lipschitz-type conditions involving a parameter $$q>1$$. He shows how changes in the position of $$q$$ with respect to $$p$$ lead to different behavior of the solution set. When dealing with sign-changing solutions, he assumes that $$f$$ is half-odd; a condition generalizing the usual oddness. A quadrature method is used.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
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