Positive solutions for a nonlinear problem involving the bosonic operator

Juan Arratia; Pedro Ubilla

doi:10.3934/dcds.2026074

Article Contents

2026, Volume 54: 473-507. Doi: 10.3934/dcds.2026074

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Positive solutions for a nonlinear problem involving the bosonic operator

Juan Arratia^1, and
Pedro Ubilla^{1, 2, ,}

1.
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
2.
Departamento de Matemática, Universidade Federal do Rio Grande do Norte, 9078–970, Natal–RN, Brazil

^*Corresponding author: Pedro Ubilla
^*Corresponding author: Pedro Ubilla

Received: February 4, 2026

Revised: March 20, 2026

Early access: April 1, 2026

Published online: April 1, 2026

The second author is supported by [FONDECYT Grant 1220675].

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Abstract

We study the existence and multiplicity of positive solutions for a class of nonlocal elliptic equations involving the bosonic operator $ -\Delta e^{-c\Delta} $, which arises in models from bosonic string theory and nonlocal cosmology.

For the parametric problem with a nonlinearity that is sublinear at infinity,

$ -\Delta e^{-c\Delta}u + u = \lambda a(x)g(u) \quad \text{in } \mathbb{R}^N, $

we use tools tailored to the nonlocal framework, including sharp decay estimates for the associated kernel, a strong maximum principle, and the method of lower and upper solutions.

Our main result identifies a critical parameter $ \lambda^*>0 $ such that the problem admits no positive solutions for $ \lambda<\lambda^* $, at least one solution for $ \lambda = \lambda^* $, and at least two ordered positive solutions for $ \lambda>\lambda^* $. One solution follows from coercivity arguments together with the method of lower and upper solutions, whereas the second is obtained through a nontrivial linking argument. This multiplicity pattern reveals that the problem exhibits a structure that can be viewed as a dual counterpart to the classical concave-convex type equations. A key ingredient in this analysis is a local minimizer equivalence principle between the natural energy space and a suitable weighted space with nonempty interior, which enables comparison and minimization techniques in the whole space.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35J20, 35B09, 45K05.

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