Volume 46, Issue 6 pp. 7454-7465
RESEARCH ARTICLE

On non-local elliptic equations with sublinear nonlinearities involving an eigenvalue problem

Ching-yu Chen

Ching-yu Chen

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811 Taiwan

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Yueh-cheng Kuo

Yueh-cheng Kuo

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811 Taiwan

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Kuan-Hsiang Wang

Kuan-Hsiang Wang

Department of Mathematics, National Cheng Kung University, Tainan, Taiwan

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Tsung-fang Wu

Corresponding Author

Tsung-fang Wu

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811 Taiwan

Correspondence

Tsung-fang Wu, Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan.

Email: [email protected]

Communicated by: P. Agarwal

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First published: 23 December 2022

Abstract

A non-local elliptic equation of Kirchhoff-type

a Ω | u | 2 d x + 1 Δ u = λ f ( x ) u + g ( x ) | u | γ 2 u in Ω $$ -\left(a{\int}_{\Omega}{\left|\nabla u\right|}^2 dx+1\right)\Delta u=\lambda f(x)u+g(x){\left|u\right|}^{\gamma -2}u\kern0.20em \mathrm{in}\kern0.60em \Omega $$
for a , λ > 0 $$ a,\lambda &gt;0 $$ with Dirichlet boundary conditions is investigated for the cases where 1 < γ < 2 $$ 1&lt;\gamma &lt;2 $$ . It is well known that with the non-local effect removed and f 1 $$ f\equiv 1 $$ , a branch of positive solutions bifurcates from infinity at λ = λ 1 $$ \lambda &#x0003D;{\lambda}_1 $$ and no positive solution exists whenever λ > λ $$ \lambda &gt;\overline{\lambda} $$ for some λ λ 1 $$ \overline{\lambda}\ge {\lambda}_1 $$ (see K. J. Brown, Calc. Var. 22, 483-494, 2005),where λ 1 $$ {\lambda}_1 $$ is the principal eigenvalue of the linear problem Δ u = λ u $$ -\Delta u&#x0003D;\lambda u $$ . As a consequence of the non-local effect, our analysis has found no bifurcation from infinity, and at least one positive solution is always permitted for λ > 0 $$ \lambda &gt;0 $$ . Moreover, regions with three positive solutions are found for small value of a $$ a $$ . Comparisons are also made of the results here with those of the elliptic problem in the absence of the non-local term under the same prescribed conditions using numerical simulations.

CONFLICT OF INTEREST

This work does not have any conflict of interest.

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