Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball

Abstract

Please cite this article in press as: R. Dhanya et al., Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball, J. Math. Anal. Appl. (2015), http://dx. AbstractWe study positive radial solutions tois a class of non-decreasing functions satisfying lim s→∞ f (s) s = ∞ (superlinear) and f (0) < 0 (semipositone). We consider solutions, u, such that u → 0 as |x| → ∞, and which also satisfy the nonlinear boundary conditon ∂u ∂η +c(u)u = 0 when |x| = r 0 , where ∂ ∂η is the outward normal derivative, andc ∈ C ([0, ∞), (0, ∞)). We will establish the existence of a positive radial solution for small values of the parameter λ. We also establish a similar result for the case when u satisfies the Dirichlet boundary conditon (u = 0) for |x| = r 0 . We establish our results via variational methods, namely using the Mountain Pass Lemma.

Existence of positive radial solutions for a superlinear semipositone p-Laplacian problem on the exterior of a ball

“…Recently, when p = 2, Dhanya et al . [13] proved the existence of a positive radial solution when Ω is the region exterior to a ball. Their study also included the case in which a nonlinear condition (as in (1.2)) was satisfied on the inner boundary (i.e.…”

“…The focus of this paper is to extend this result for all p > 1. In [13], the Dhanya et al . used variational methods (the mountain pass theorem) combined with the properties of the Green function.…”

Non-Zero Radial Solutions for Elliptic Systems with Coupled Functional BCs in Exterior Domains

“…In the context of non-homogeneous BCs, elliptic problems in exterior domains were studied by Aftalion and Busca [2] and doÓ et al [13][14][15][16], and nonlinear BCs were investigated by Butler and others [4], Dhanya et al [9], Ko and co-authors [33], and Lee and others [40].…”

Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions

“…In particular, our existence result (Theorem 1.1) improves and extends (to the infinite semipositone case) a corresponding result in [15,Theorem 1.2], in which the existence of a positive solution to (1.2) was established for λ > 0 small when f : [0, ∞) → R is continuous, nondecreasing with f (0) < 0, A(s q − 1) ≤ f (s) ≤ B(s q + 1) for some q > p − 1, A, B > 0, and there exists a constant θ > p such that sf (s) > θ s 0 f and c(s) < θg(s)/s p for s large. The results in [15], which provide an extension of the ones in [9] to the p-Laplace case, were established via the Mountain Pass Lemma in variational methods. Our approach here makes use of degree theory and comparison principles, which help avoiding the technical assumptions associated with variational methods in [9,15].…”

“…The results in [15], which provide an extension of the ones in [9] to the p-Laplace case, were established via the Mountain Pass Lemma in variational methods. Our approach here makes use of degree theory and comparison principles, which help avoiding the technical assumptions associated with variational methods in [9,15]. In particular, our results when applied to the model case…”