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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 & 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.   Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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