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2015, 2015(special): 436-445. doi: 10.3934/proc.2015.0436

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

Guglielmo Feltrin 1,

1. 

SISSA - International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy

Received  September 2014 Revised  September 2015 Published  November 2015

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.
Keywords: existence result., indefi nite weight, boundary value problem, superlinear equation, Positive solution.
Mathematics Subject Classification: Primary: 34B18; Secondary: 34B1.
Citation: Guglielmo Feltrin. Existence of positive solutions of a superlinear boundary value problem with indefinite weight. Conference Publications, 2015, 2015 (special) : 436-445. doi: 10.3934/proc.2015.0436
References:
[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.

show all references

References:
[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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