Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions

We prove new results on the existence, non-existence, localization and multiplicity of nontrivial radial solutions of a system of elliptic boundary value problems on exterior domains subject to nonlocal, nonlinear, functional boundary conditions. Our approach relies on fixed point index theory. As a by-product of our theory we provide ananswer to an open question posed by doÓ, Lorca, Sánchez and Ubilla. We include some examples with explicit nonlinearities in order to illustrate our theory.

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“…In order to establish the existence of radial solutions w = w(r), r = |x|, we proceed as in [4] and rewrite (2.2) in the form…”

Section: A System Of Elliptic Pdes In Exterior Domainsmentioning

confidence: 99%

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

Cianciaruso¹,

Infante²,

Pietramala³

2019

Proceedings of the Edinburgh Mathematical Society

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“…Here, the authors study a model arising in combustion theory involving a sublinear reaction term. See also [9] for an extension of this work to an exterior domain.…”

Section: Introductionmentioning

confidence: 97%

“…Note that the change of variables r = |x| and S = r r0 2−N transforms (1), (2) respectively into the following boundary value problems (see Appendix 8.1 in [9] for details):…”

Section: Introductionmentioning

confidence: 99%

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

Dhanya

Morris

Shivaji

2016

Journal of Mathematical Analysis and Applications

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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.

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“…Such nonlinear boundary conditions occur very naturally in applications (see [12] for a detailed description in a model arising in combustion theory). Recently, the existence of a radial positive solution for (1.1) when λ 1 has been established in [5], via the method of sub-super solutions. Here we discuss the uniqueness of this radial solution when some additional assumptions hold.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the change of variable r = |x| and s = ( r r 0 ) 2−N transforms (1.1) into the following boundary value problem (see Appendix 8.1 in [5] for details):…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball

Ramaswamy

Shivaji

2015

Journal of Mathematical Analysis and Applications

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We study positive radial solutions to: −Δu = λK (|x|)= 0 and f (0) < 0 (semipositone). We are interested in solutions u such that u → 0 as |x| → ∞ and satisfy the nonlinear boundary condition ∂u ∂η +c(u)u = 0 if |x| = r 0 where ∂ ∂η is the outward normal derivative and c ∈ C([0, ∞), (0, ∞)) is an increasing function. We will establish the uniqueness of positive radial solutions for large values of the parameter λ.

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Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions

Cited by 17 publications

References 17 publications

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

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

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

Uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball

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