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
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.   Google Scholar

[2]

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

[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.   Google Scholar

[4]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems,, in Differential equations (São Paulo, (1981), 34.   Google Scholar

[5]

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

[6]

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

[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.   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

A. Zettl, Sturm-Liouville theory,, Mathematical Surveys and Monographs, (2005).   Google Scholar

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.   Google Scholar

[2]

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

[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.   Google Scholar

[4]

D. G. de Figueiredo, Positive solutions of semilinear elliptic problems,, in Differential equations (São Paulo, (1981), 34.   Google Scholar

[5]

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

[6]

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

[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.   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

A. Zettl, Sturm-Liouville theory,, Mathematical Surveys and Monographs, (2005).   Google Scholar

[1]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[2]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513

[3]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[4]

Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl. The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 151-196. doi: 10.3934/dcds.2010.26.151

[5]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[6]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[7]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[8]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[9]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73

[10]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021

[11]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[12]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[13]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[14]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[15]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[16]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[17]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[18]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[19]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[20]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

 Impact Factor: 

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]