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

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

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.
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
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show all references

References:
[1]

J. Differential Equations, 146 (1998), 336-374. Google Scholar

[2]

Topol. Methods Nonlinear Anal., 4 (1994), 59-78. Google Scholar

[3]

J. Differential Equations, 214 (2005), 36-64. Google Scholar

[4]

in Differential equations (São Paulo, 1981), vol. 957 of Lecture Notes in Math., Springer, Berlin-New York, 1982, pp. 34-87 Google Scholar

[5]

Proc. Amer. Math. Soc., 120 (1994), 743-748. Google Scholar

[6]

J. Differential Equations, 259 (2015), 925-963. Google Scholar

[7]

Commun. Pure Appl. Anal., 2 (2003), 411-423. Google Scholar

[8]

J. Differential Equations, 148 (1998), 407-421. Google Scholar

[9]

Differential Integral Equations, 8 (1995), 213-222. Google Scholar

[10]

J. Differential Equations, 14 (1973), 360-394. Google Scholar

[11]

J. Math. Anal. Appl., 51 (1975), 461-482. Google Scholar

[12]

Mathematical Surveys and Monographs, 121, American Mathematical Society, Providence, RI, 2005. Google Scholar

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