Positive radial solutions of a quasilinear problem in an exterior domain with vanishing boundary conditions
DOI:
https://doi.org/10.12775/TMNA.2020.050Keywords
Quasi-linear boundary value problem, exterior domain, positive radial solutions, Krasnosel'skii fixed point theoremAbstract
In this work, we study the existence and nonexistence of positive radial solutions for the quasilinear equation $\mathrm{div}(A(|\nabla u|)\nabla u)+\lambda k(|x|)f(u)=0$ in the exterior of a ball with vanishing boundary conditions using an approach based on a fixed point theorem for operators on Banach Space.References
C. Bandle, C.V. Coffman and M. Marcus, Nonlinear elliptic problems in annular domains, J. Differential Equations 69 (1987), 322–345.
C. Bandle and M.K. Kwong, Semilinear elliptic problems in annular domains, J. Appl. Math. Phys. (ZAMP) 40 (1989), 245–257.
M. Chhetri, P. Drábek and R. Shivaji, Analysis of positive solutions for classes of quasilinear singular problems on exterior domains, Adv. Nonlinear Anal. 6 (2017), 447–459.
M. Chhetri, P. Drábek and R. Shivaji, S-shaped bifurcation diagrams in exterior domains, Positivity 23 (2019), 1147–1164.
S. Coleman, V. Glazer and A. Martin, Action minima among solutions to a class of Euclidean scalar field equations, Comm. Math. Phys. 58 (1978), 211–221.
C.V. Coffman and M. Marcus, Existenceand uniqueness results for semilinear Dirichlet problems in annuli, Arch. Rational Mech. Anal. 108 (1989), 293–307.
K. Deimling, Nonlinear Functional Analysis, Springer, 1995.
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.
J.M. do O, S. Lorca, J. Sánchez and P. Ubilla, Non-homogeneous elliptic equations in exterior domain, Proc. Roy. Soc. Edinburgh Ser. A 136 (2006), 139–147.
I.M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Tansl. Ser. 2 29 (1963), 295–381.
D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, 1988.
D.D. Hai and R. Shivaji, Positive radial solutions of a class os singular superlinear problems on the exterior of a ball with nonlinear boundary conditions, J. Math. Anal. Appl. 456 (2017), 872–881.
D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1973), 241–269.
L. Kong and J. Wang, Multiple positive solutions for the one-dimensional p-Laplacian, Nonlinear Anal. 42 (2000), 1327–1333.
M.A. Krasnosel’skiı̆, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
S.S. Lin, On the existence of positive radial solutions for semilinear elliptic equations in annular domains, J. Differential Equations 81 (1989), 221–233.
W.M. Ni and J. Serrin, Non-existence theorems for singular solutions of quasilinear partial differential equations, Comm. Pure Appl. Math. 39 (1986), 379–399.
W.M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo Suppl. 5 (1986), 171–185.
J. Sánchez, Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional p-laplacian, J. Math. Anal. Appl. 292 (2004), 401–414.
R. Stańczy, Positive solutions for superlinear elliptic equations, J. Math. Anal. Appl. 283 (2003), 159–166.
H. Wang, On the existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 109 (1994), 1–7.
J. Wang, The existence of positive solutions for the one-dimensional p-Laplacian, Proc. Amer. Math. Soc. 125 (1997), 2275–2283.
H.Wang, On the structure of positive radial solutions for quasilinear equations in annular domains, Adv. Differential Equations 8 (2003), 111–128.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0