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September  2020, 19(9): 4655-4666. doi: 10.3934/cpaa.2020131

Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions

1. 

Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA

2. 

Department of Mathematics and Statisitics, University of North Carolina at Greensboro, Greensboro, NC 27402, USA

* Corresponding author

Received  March 2019 Revised  November 2019 Published  June 2020

We prove the existence of positive radial solutions to the problem
$ \begin{cases} -\Delta _{p}u = \lambda \ K(|x|)f(u)\ \text{in } |x|>r_{0}, \\ \dfrac{\partial u}{\partial n}+\tilde{c}(u)u = 0\ \text{on }|x| = r_{0},\ \ u(x)\rightarrow 0\text{ as }|x|\rightarrow \infty ,\end{cases} $
where
$ \ \Delta _{p}u = div(|\nabla u|^{p-2}\nabla u),\ N>p>1, \Omega = \{x\in \mathbb{R}^{N}:|x|>r_{0}>0\}, $
$ f:(0,\infty )\rightarrow \mathbb{R} $
is
$ p $
-superlinear at
$ \infty $
with possible singularity at
$ 0, $
and
$ \lambda $
is a small positive parameter. A nonexistence result is also established when
$ f $
has semipositone structure at
$ 0. $
Citation: Trad Alotaibi, D. D. Hai, R. Shivaji. Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4655-4666. doi: 10.3934/cpaa.2020131
References:
[1]

W. AllegrettoP. Nistri and P. Zecca, Positive solutions of elliptic nonpositone problems, Differ. Integral Equ., 5 (1992), 95-101. 

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, Siam Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.

[3]

A. AmbrosettiD. Arcoya and B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory, Differ. Integral Equ., 7 (1994), 655-663. 

[4]

V. AnuradhaD. D. Hai and R. Shivaji, Existence results for superlinear semipositone BVP's, Proc. Amer. Math. Soc., 124 (1996), 757-763.  doi: 10.1090/S0002-9939-96-03256-X.

[5]

D. Arcoya and A. Zertiti, Existence and non-existence of radially symmetric non-negative solutions for a class of semipositone problems in an annulus, Rend. Math. Appl., 14 (1994), 625-646. 

[6]

H. BerestyckiL. Caffarelli and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains, Duke Math. J., (1996), 467-494.  doi: 10.1215/S0012-7094-96-08117-X.

[7]

K. J. BrownA. Castro and R. Shivaji, Nonexistence of radially symmetric nonnegative solutionsfor a class of semipositone problems, Differ. Integral Equ., 2 (1989), 541-545. 

[8]

M. Chhetri and P. Girg, Existence of positive solutions for a class of superlinear semipositone systems, J. Math. Anal. Appl., 408 (2013), 781-788.  doi: 10.1016/j.jmaa.2013.06.041.

[9]

R. DhanyaQ. Morris and R. Shivaji, Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 434 (2016), 1533-1548.  doi: 10.1016/j.jmaa.2015.07.016.

[10]

D. D. Hai, On singular Sturm-Liouville boundary value problems, Proc. R. Soc. Edinb., 140A (2010), 49-63.  doi: 10.1017/S0308210508000358.

[11]

D. D. Hai and R. Shivaji, Positive radial solutions for a class of singular superlinear problems on the exterior of a ball with nonlinear boundary conditions, J. Math. Anal. Appl., 456 (2017), 872-881.  doi: 10.1016/j.jmaa.2017.06.088.

[12]

J. Jacobsen and K. Schmitt, Radial solutions of quasilinear elliptic differential equations, in Handbook of Differential Equations, Elsevier/North-Holand, Amsterdam, (2004) 359–435.

[13]

E. KoM. Ramaswasmy and R. Shivaji, Uniqueness of positive solutions for a class of semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 423 (2015), 399-409.  doi: 10.1016/j.jmaa.2014.09.058.

[14]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.

[15]

Q. MorrisR. Shivaji and I. Sim, Existence of positive radial solutions for a superlinear semipositone $p$-Laplacian problem on the exterior of a ball, Proc. R. Soc. Edinb., 148A (2018), 409-428.  doi: 10.1017/S0308210517000452.

[16]

R. ShivajiI. Sim and B. Son, A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball, J. Math. Anal. Appl., 445 (2017), 459-475.  doi: 10.1016/j.jmaa.2016.07.029.

[17]

J. Smoller and A. Wasserman, Existence of positive solutions for semilinear elliptic equations in general domains, Arch. Ration. Mech. Anal., 98 (1987), 229-249.  doi: 10.1007/BF00251173.

show all references

References:
[1]

W. AllegrettoP. Nistri and P. Zecca, Positive solutions of elliptic nonpositone problems, Differ. Integral Equ., 5 (1992), 95-101. 

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, Siam Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.

[3]

A. AmbrosettiD. Arcoya and B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory, Differ. Integral Equ., 7 (1994), 655-663. 

[4]

V. AnuradhaD. D. Hai and R. Shivaji, Existence results for superlinear semipositone BVP's, Proc. Amer. Math. Soc., 124 (1996), 757-763.  doi: 10.1090/S0002-9939-96-03256-X.

[5]

D. Arcoya and A. Zertiti, Existence and non-existence of radially symmetric non-negative solutions for a class of semipositone problems in an annulus, Rend. Math. Appl., 14 (1994), 625-646. 

[6]

H. BerestyckiL. Caffarelli and L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains, Duke Math. J., (1996), 467-494.  doi: 10.1215/S0012-7094-96-08117-X.

[7]

K. J. BrownA. Castro and R. Shivaji, Nonexistence of radially symmetric nonnegative solutionsfor a class of semipositone problems, Differ. Integral Equ., 2 (1989), 541-545. 

[8]

M. Chhetri and P. Girg, Existence of positive solutions for a class of superlinear semipositone systems, J. Math. Anal. Appl., 408 (2013), 781-788.  doi: 10.1016/j.jmaa.2013.06.041.

[9]

R. DhanyaQ. Morris and R. Shivaji, Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 434 (2016), 1533-1548.  doi: 10.1016/j.jmaa.2015.07.016.

[10]

D. D. Hai, On singular Sturm-Liouville boundary value problems, Proc. R. Soc. Edinb., 140A (2010), 49-63.  doi: 10.1017/S0308210508000358.

[11]

D. D. Hai and R. Shivaji, Positive radial solutions for a class of singular superlinear problems on the exterior of a ball with nonlinear boundary conditions, J. Math. Anal. Appl., 456 (2017), 872-881.  doi: 10.1016/j.jmaa.2017.06.088.

[12]

J. Jacobsen and K. Schmitt, Radial solutions of quasilinear elliptic differential equations, in Handbook of Differential Equations, Elsevier/North-Holand, Amsterdam, (2004) 359–435.

[13]

E. KoM. Ramaswasmy and R. Shivaji, Uniqueness of positive solutions for a class of semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 423 (2015), 399-409.  doi: 10.1016/j.jmaa.2014.09.058.

[14]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.

[15]

Q. MorrisR. Shivaji and I. Sim, Existence of positive radial solutions for a superlinear semipositone $p$-Laplacian problem on the exterior of a ball, Proc. R. Soc. Edinb., 148A (2018), 409-428.  doi: 10.1017/S0308210517000452.

[16]

R. ShivajiI. Sim and B. Son, A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball, J. Math. Anal. Appl., 445 (2017), 459-475.  doi: 10.1016/j.jmaa.2016.07.029.

[17]

J. Smoller and A. Wasserman, Existence of positive solutions for semilinear elliptic equations in general domains, Arch. Ration. Mech. Anal., 98 (1987), 229-249.  doi: 10.1007/BF00251173.

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