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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:
  W. Allegretto, P. Nistri and P. Zecca, Positive solutions of elliptic nonpositone problems, Differ. Integral Equ., 5 (1992), 95-101.  H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, Siam Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.   A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory, Differ. Integral Equ., 7 (1994), 655-663.  V. Anuradha, D. 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.   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.  H. Berestycki, L. 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.   K. J. Brown, A. Castro and R. Shivaji, Nonexistence of radially symmetric nonnegative solutionsfor a class of semipositone problems, Differ. Integral Equ., 2 (1989), 541-545.  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.   R. Dhanya, Q. 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.   D. D. Hai, On singular Sturm-Liouville boundary value problems, Proc. R. Soc. Edinb., 140A (2010), 49-63.  doi: 10.1017/S0308210508000358.   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.   J. Jacobsen and K. Schmitt, Radial solutions of quasilinear elliptic differential equations, in Handbook of Differential Equations, Elsevier/North-Holand, Amsterdam, (2004) 359–435.  E. Ko, M. 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.   P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.   Q. Morris, R. 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.   R. Shivaji, I. 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.   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:
  W. Allegretto, P. Nistri and P. Zecca, Positive solutions of elliptic nonpositone problems, Differ. Integral Equ., 5 (1992), 95-101.  H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, Siam Rev., 18 (1976), 620-709.  doi: 10.1137/1018114.   A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semipositone problems via bifurcation theory, Differ. Integral Equ., 7 (1994), 655-663.  V. Anuradha, D. 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.   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.  H. Berestycki, L. 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.   K. J. Brown, A. Castro and R. Shivaji, Nonexistence of radially symmetric nonnegative solutionsfor a class of semipositone problems, Differ. Integral Equ., 2 (1989), 541-545.  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.   R. Dhanya, Q. 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.   D. D. Hai, On singular Sturm-Liouville boundary value problems, Proc. R. Soc. Edinb., 140A (2010), 49-63.  doi: 10.1017/S0308210508000358.   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.   J. Jacobsen and K. Schmitt, Radial solutions of quasilinear elliptic differential equations, in Handbook of Differential Equations, Elsevier/North-Holand, Amsterdam, (2004) 359–435.  E. Ko, M. 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.   P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.   Q. Morris, R. 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.   R. Shivaji, I. Sim and B. Son, A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball, J. Math. Anal. 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