On oscillating radial solutions for non-autonomous semilinear elliptic equations

H. Al Jebawy; H. Ibrahim; Z. Salloum; H. Al Jebawy; H. Ibrahim; Z. Salloum

doi:10.3934/math.2024737

AIMS Mathematics

2024, Volume 9, Issue 6: 15190-15201. doi: 10.3934/math.2024737

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Research article

On oscillating radial solutions for non-autonomous semilinear elliptic equations

1.
American University in Dubai (AUD), Dubai, UAE
2.
Faculty of Science I, Mathematics Department, Lebanese University, Hadath, Lebanon

Received: 18 February 2024 Revised: 03 April 2024 Accepted: 08 April 2024 Published: 26 April 2024
MSC : 34C10, 35J61

We consider semilinear elliptic equations of the form $ \Delta u + f(|x|, u) = 0 $ on $ {\mathbb{R}}^{N} $ with $ f(|x|, u) = q(|x|)g(u) $. These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity $ f $, there exists an oscillating radial solution converging to a zero of the function $ g $. We also study the oscillating and limiting behavior of this solution.
- semilnear elliptic equation,
- oscillation theory,
- positive radial solution
Citation: H. Al Jebawy, H. Ibrahim, Z. Salloum. On oscillating radial solutions for non-autonomous semilinear elliptic equations[J]. AIMS Mathematics, 2024, 9(6): 15190-15201. doi: 10.3934/math.2024737

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Abstract

We consider semilinear elliptic equations of the form $ \Delta u + f(|x|, u) = 0 $ on $ {\mathbb{R}}^{N} $ with $ f(|x|, u) = q(|x|)g(u) $. These type of equations arise in various problems in applied mathematics, and particularly in the study of population dynamics, solitary waves, diffusion processes, and phase transitions. We show that under suitable assumptions on the nonlinearity $ f $, there exists an oscillating radial solution converging to a zero of the function $ g $. We also study the oscillating and limiting behavior of this solution.

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