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Infinite multiplicity of stable entire solutions for a semilinear elliptic equation with exponential nonlinearity

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  17 January 2019

Soohyun Bae
Soohyun Bae*
Affiliation:
Department of Mathematical Sciences, Hanbat National University Daejeon 34158, Republic of Korea (shbae@hanbat.ac.kr)
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Abstract

We consider the infinite multiplicity of entire solutions for the elliptic equation Δu + K(x)eu + μf(x) = 0 in ℝn, n ⩾ 3. Under suitable conditions on K and f, the equation with small μ ⩾ 0 possesses a continuum of entire solutions with a specific asymptotic behaviour. Typically, K behaves like |x|ℓ at ∞ for some ℓ > −2 and the entire solutions behave asymptotically like − (2 + ℓ)log |x| near ∞. Main tools of the analysis are comparison principle for separation structure, asymptotic expansion of solutions near ∞, barrier method and strong maximum principle. The linearized operator for the equation has two characteristic behaviours related with the stability and the weak asymptotic stability of the solutions as steady states for the corresponding parabolic equation.

Keywords

semilinear elliptic equationexponential nonlinearitystable entire solutionseparationinfinite multiplicity

MSC classification

Primary: 35J61: Semilinear elliptic equations
Secondary: 35B08: Entire solutions 35B35: Stability 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1371 - 1404
Copyright
Copyright © Royal Society of Edinburgh 2019 

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