Positive radial solutions for a class of singular superlinear problems on the exterior of a ball with nonlinear boundary conditions. (English) Zbl 1392.35110
Summary: We discuss existence and nonexistence results of positive radial solutions to the problem
\[
\begin{cases} \begin{gathered} - {\Delta} u = \lambda K(| x |) f(u)\;\text{in } | x | > r_0, \\ \frac{\partial u}{\partial n} + \widetilde{c}(u) u = 0 \text{ on } | x | = r_0,\quad u(x) \rightarrow 0 \text{ as } | x | \rightarrow \infty, \end{gathered}\end{cases}
\]
where \(\Omega =\big\{x \in\mathbb R^N : | x | > r_0 > 0 \big\}\), \(N > 2\), \(f:(0, \infty) \rightarrow \mathbb{R}\) is continuous, superlinear at \(\infty\), and is allowed to be singular at 0 with no sign conditions near 0.
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
singular superlinear problems; nonlinear boundary conditions; exterior domains; positive radial solutionsReferences:
| [1] | Ali, I.; Castro, A.; Shivaji, R., Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc., 117, 775-782 (1993) · Zbl 0770.34019 |
| [2] | Allegretto, W.; Nistri, P.; Zecca, P., Positive solutions of elliptic nonpositone problems, Differential Integral Equations, 5, 95-101 (1992) · Zbl 0758.35032 |
| [3] | Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044 |
| [4] | Ambrosetti, A.; Arcoya, D.; Buffoni, B., Positive solutions for some semipositone problems via bifurcation theory, Differential Integral Equations, 7, 655-663 (1994) · Zbl 0808.35030 |
| [5] | Anuradha, V.; Hai, D. D.; Shivaji, R., Existence results for superlinear semipositone BVP’s, Proc. Amer. Math. Soc., 124, 757-763 (1996) · Zbl 0857.34032 |
| [6] | Berestycki, H.; Caffarelly, L.; Nirenberg, L., Inequalities for second-order elliptic equations with applications to unbounded domains, Duke Math. J., 467-494 (1996) · Zbl 0860.35004 |
| [7] | Brown, K. J.; Castro, A.; Shivaji, R., Nonexistence of radially symmetric nonnegative solutions for a class of semipositone problems, Differential Integral Equations, 2, 541-545 (1989) · Zbl 0736.35039 |
| [8] | Castro, A.; Gadam, S.; Shivaji, R., Branches of radial solutions for semipositone problems, J. Differential Equations, 120, 30-45 (1995) · Zbl 0838.34037 |
| [9] | Castro, A.; Shivaji, R., Non-negative solutions for a class of radially symmetric non-positone problems, Proc. Amer. Math. Soc., 106, 735-740 (1989) · Zbl 0686.35044 |
| [10] | Castro, A.; Shivaji, R., Non-negative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric, Comm. Partial Differential Equations, 14, 1091-1100 (1989) · Zbl 0688.35025 |
| [11] | Dhanya, R.; Morris, Q.; Shivaji, R., Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 434, 1533-1548 (2016) · Zbl 1328.35024 |
| [12] | Hai, D. D., On singular Sturm-Liouville boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 140, 49-63 (2010) · Zbl 1194.34038 |
| [13] | Hai, D. D., Positive radial solutions for singular quasilinear elliptic equations in a ball, Publ. Res. Inst. Math. Sci., 50, 2, 341-362 (2014) · Zbl 1303.35038 |
| [14] | Hai, D. D.; Shivaji, R., On radial solutions for singular combined superlinear elliptic systems on annular domains, J. Math. Anal. Appl., 446, 1, 335-344 (2017) · Zbl 1391.34055 |
| [15] | Jacobsen, J.; Schmitt, K., Radial solutions of quasilinear elliptic differential equations, (Handbook of Differential Equations (2004), Elsevier/North-Holand: Elsevier/North-Holand Amsterdam), 359-435 · Zbl 1091.34011 |
| [16] | Ko, E.; Ramaswasmy, M.; Shivaji, R., Uniqueness of positive solutions for a class of semipositone problems on the exterior of a ball, J. Math. Anal. Appl., 423, 399-409 (2015) · Zbl 1308.35107 |
| [17] | Lions, P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24, 441-467 (1982) · Zbl 0511.35033 |
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