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Existence of positive solutions of a superlinear boundary value problem with indefinite weight

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  • We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change sign. We assume that the function $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, including the superlinear case $g(s)=s^{p}$, with $p>1$. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.
    Mathematics Subject Classification: Primary: 34B18; Secondary: 34B15.

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  • [1]

    H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.

    [2]

    H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.

    [3]

    D. Bonheure, J. M. Gomes and P. Habets, Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.

    [4]

    D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, in Differential equations (São Paulo, 1981), vol. 957 of Lecture Notes in Math., Springer, Berlin-New York, 1982, pp. 34-87

    [5]

    L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743-748.

    [6]

    G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: a topological approach, J. Differential Equations, 259 (2015), 925-963.

    [7]

    M. Gaudenzi, P. Habets and F. Zanolin, Positive solutions of superlinear boundary value problems with singular indefinite weight, Commun. Pure Appl. Anal., 2 (2003), 411-423.

    [8]

    K. Lan and J. R. L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148 (1998), 407-421.

    [9]

    R. Manásevich, F. I. Njoku and F. Zanolin, Positive solutions for the one-dimensional $p$-Laplacian, Differential Integral Equations, 8 (1995), 213-222.

    [10]

    R. D. Nussbaum, Periodic solutions of some nonlinear, autonomous functional differential equations. II, J. Differential Equations, 14 (1973), 360-394.

    [11]

    R. D. Nussbaum, Positive solutions of nonlinear elliptic boundary value problems, J. Math. Anal. Appl., 51 (1975), 461-482.

    [12]

    A. Zettl, Sturm-Liouville theory, Mathematical Surveys and Monographs, 121, American Mathematical Society, Providence, RI, 2005.

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