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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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