# zbMATH — the first resource for mathematics

Uniqueness and stability of nonnegative solutions for semipositone problems in a ball. (English) Zbl 0770.34019
If $$f:[0,\infty)\to\mathbb{R}$$ is monotonically increasing, $$f(0)<0$$, $$f(u)>0$$ for some $$u>0$$ and $$f$$ is concave, then there exists $$\mu_ 1$$ such that if $$\lambda>\mu_ 1$$, then the problem $$\Delta u+\lambda f(u)=0$$ in $$\Omega$$, $$\Omega$$ is the unit ball in $$\mathbb{R}^ n$$, $$n\geq 2$$, $$u=0$$ on $$\partial\Omega$$, has at most one nonnegative solution. Furthermore, there exists $$\mu_ 2$$ such that all nonnegative solutions for $$\lambda>\mu_ 2$$ are stable. A similar result is obtained if $$f$$ is convex and some other conditions hold.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text:
##### References:
 [1] K. J. Brown, Alfonso Castro, and R. Shivaji, Nonexistence of radially symmetric nonnegative solutions for a class of semi-positone problems, Differential Integral Equations 2 (1989), no. 4, 541 – 545. · Zbl 0736.35039 [2] K. J. Brown and R. Shivaji, Instability of nonnegative solutions for a class of semipositone problems, Proc. Amer. Math. Soc. 112 (1991), no. 1, 121 – 124. · Zbl 0741.35003 [3] A. Castro, J. B. Garner, and R. Shivaji, Existence results for a class of sublinear semipositone problems, Result. Math. (to appear). · Zbl 0785.35073 [4] Alfonso Castro and R. Shivaji, Nonnegative solutions for a class of nonpositone problems, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 3-4, 291 – 302. · Zbl 0659.34018 [5] Alfonso Castro and R. Shivaji, Nonnegative solutions for a class of radially symmetric nonpositone problems, Proc. Amer. Math. Soc. 106 (1989), no. 3, 735 – 740. · Zbl 0686.35044 [6] Alfonso Castro and R. Shivaji, Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric, Comm. Partial Differential Equations 14 (1989), no. 8-9, 1091 – 1100. · Zbl 0688.35025 [7] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209 – 243. · Zbl 0425.35020 [8] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 979 – 1000. · Zbl 0223.35038 [9] Joel Smoller and Arthur Wasserman, Existence of positive solutions for semilinear elliptic equations in general domains, Arch. Rational Mech. Anal. 98 (1987), no. 3, 229 – 249. · Zbl 0664.35029 [10] G. Sweers, Semilinear elliptic eigenvalue problems, Doctoral Thesis, Univ. of Leiden, Netherlands, 1988. · Zbl 0681.35013 [11] S. Unsurangsie, Existence of a solution for a wave equation and an elliptic Dirichlet problem, Doctoral Thesis, Univ. of North Texas, 1988. · Zbl 0699.35176
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.