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**Positive solutions for the 1-dimensional generalized \(p\)-Laplacian involving a real parameter.**
*(English)*
Zbl 1082.34511

From the introduction: We study positive solutions of the Dirichlet problem \((\text{P})_\lambda: -(\phi(u'))'=\lambda f(u)\) in \((0,1), u(0)=u(1)=0\), where \(\lambda\in\mathbb R, \phi: \mathbb R\to\mathbb R\) is an increasing odd homeomorphism and \(f\: \mathbb R\to\mathbb R\) is continuous. By a solution of problem \(\text{P}_\lambda\) we understand a function \(u\in C^1(0,1)\) such that \(\phi(u')\in C^1(0,1)\) satisfying the equations in \(\text{P}_\lambda\).

For the partial differential case, the existence of positive solutions for the problem \(-\Delta u=\lambda f(u)\) in \(\Omega, u=0\) on \(\partial\Omega\), has been widely studied by many authors. An informative survey concerning this problem can be found in the paper of P.-L. Lions [SIAM Rev. 24, No. 4, 441–467 (1982; Zbl 0511.35033)], who considered different behaviors of \(f\) in order to obtain existence and multiplicity results of positive solutions. On the other hand, in the case where \(f\) is superlinear and the partial differential operator is asymptotic to a power at zero and at infinity, K. Narukawa and T. Suzuki, Funkc. Ekvacioj, Ser. Int. 37, No. 1, 81–100 (1994; Zbl 0807.35044)] studied the existence of positive solutions, and obtained the existence of a \(\lambda^*>0\) such that for any \(0<\lambda<\lambda^*\) the problem described has at least two positive solutions; nevertheless the existence of positive solutions for \(\lambda\geq\lambda^*\) was not considered. In the 1-dimensional case it is also possible to obtain existence results for \(\lambda\geq\lambda^*\); thus for an operator satisfying the conditions \((A_0), (A_\infty)\) and for the nonlinearity satisfying appropriate jumping conditions (see (1.2)) we study the existence of positive solutions.

In particular, in our main result (Theorem 3.2), besides proving that \(\text{P}_\lambda\) has at least two positive solutions for any \(0<\lambda<\lambda^*\), we prove that it has at least one positive solution for \(\lambda=\lambda^*\), and has no positive solution for any \(\lambda>\lambda^*\). To this end, first we construct the solutions of \(\text{P}_\lambda\) using the shooting method (see P. Ubilla [J. Math. Anal. Appl. 190, No. 2, 611–623 (1995; Zbl 0831.34032)], and M. Guedda and L. Véron [Trans. Am. Math. Soc. 310, No. 1, 419–431 (1988; Zbl 0713.34049)]), and briefly review and refine some results found in [M. Garcia-Huidobro and P. Ubilla, Nonlinear Anal., Theory Methods Appl. 28, No. 9, 1509–1520 (1997; Zbl 0874.34021)] to obtain Lemma 2.1. Then we formulate and prove our results, and finally we present some examples.

For the partial differential case, the existence of positive solutions for the problem \(-\Delta u=\lambda f(u)\) in \(\Omega, u=0\) on \(\partial\Omega\), has been widely studied by many authors. An informative survey concerning this problem can be found in the paper of P.-L. Lions [SIAM Rev. 24, No. 4, 441–467 (1982; Zbl 0511.35033)], who considered different behaviors of \(f\) in order to obtain existence and multiplicity results of positive solutions. On the other hand, in the case where \(f\) is superlinear and the partial differential operator is asymptotic to a power at zero and at infinity, K. Narukawa and T. Suzuki, Funkc. Ekvacioj, Ser. Int. 37, No. 1, 81–100 (1994; Zbl 0807.35044)] studied the existence of positive solutions, and obtained the existence of a \(\lambda^*>0\) such that for any \(0<\lambda<\lambda^*\) the problem described has at least two positive solutions; nevertheless the existence of positive solutions for \(\lambda\geq\lambda^*\) was not considered. In the 1-dimensional case it is also possible to obtain existence results for \(\lambda\geq\lambda^*\); thus for an operator satisfying the conditions \((A_0), (A_\infty)\) and for the nonlinearity satisfying appropriate jumping conditions (see (1.2)) we study the existence of positive solutions.

In particular, in our main result (Theorem 3.2), besides proving that \(\text{P}_\lambda\) has at least two positive solutions for any \(0<\lambda<\lambda^*\), we prove that it has at least one positive solution for \(\lambda=\lambda^*\), and has no positive solution for any \(\lambda>\lambda^*\). To this end, first we construct the solutions of \(\text{P}_\lambda\) using the shooting method (see P. Ubilla [J. Math. Anal. Appl. 190, No. 2, 611–623 (1995; Zbl 0831.34032)], and M. Guedda and L. Véron [Trans. Am. Math. Soc. 310, No. 1, 419–431 (1988; Zbl 0713.34049)]), and briefly review and refine some results found in [M. Garcia-Huidobro and P. Ubilla, Nonlinear Anal., Theory Methods Appl. 28, No. 9, 1509–1520 (1997; Zbl 0874.34021)] to obtain Lemma 2.1. Then we formulate and prove our results, and finally we present some examples.