Positive solutions for sublinear elliptic equations

Abstract

Abstract. The existence of a positive radial solution for a sublinear elliptic boundary value problem in an exterior domain is proved, by the use of a cone compression fixed point theorem. The existence of a nonradial, positive solution for the corresponding nonradial problem is obtained by the sub-and supersolution method, under an additional monotonicity assumption.

On the existence of positive radial solutions for a certain class of elliptic BVPs

“…on (0, 1) and the boundedness of {p m } m∈N in L 1 (0, 1) norm and further, by (20), the boundedness of {p m } m∈N in L 2 (0, 1). Finally {p m } m∈N (up to a subsequence) tends uniformly to a certainp ∈ C((0, 1)) such thatp (t) = −g(t,x(t)) on (0, 1) and, in consequence,p ∈ C 1 ((0, 1)).…”

“…The general case, when f has not necessary the special form (variables are not separated) has been discussed, among others, in [20] (for sublinear problem), [24,25] (for superlinear problem). These results are based on topological methods.…”

“…These results are based on topological methods. The main tools used by the authors are associated with the fixed point theorems in cones due to Krasnoselskii [10, 2.3.4] (in [20,25]) and perturbation method together with some fixed point theorem which follows from Leray-Schauder degree theory [24]. The above existence results for classical solutions of this problem are obtained under the assumption that f : (1, +∞) × [0, +∞) → [0, +∞) is a continuous function satisfying some additional conditions, for example, in [24] the author assumed that Basing ourselves on methods of calculus of variations we obtain the solvability of (2) and some numerical results.…”

Nonlinear BVPS with functional parameters

“…[3,4,9,10,14,15,24] and references therein). The main tools used by the authors are based on the fixed point theorems in cones due to Krasnoselskii (in [23,26]) and perturbation method together with some fixed point theorems which follow from Leray-Schauder degree theory [25]. The authors required, among others, that the nonlinearity f : (1, +∞) × [0, +∞) → [0, +∞) is continuous and satisfies some extra conditions concerning its behavior at zero and/or at infinity.…”

Solutions of Nonlinear Singular Boundary Value Problems

“…In the case γ < 2 the Dirichlet problem (1.1)-(1.2) was investigated by several authors ( [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]). By a solution u of (1.1)-(1.2) in the continuous case we mean a function u ∈ C([0, 1], R k ) ∩ C 2 ((0, 1), R k ) satisfying (1.1) everywhere and (1.2).…”