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Existence of positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ l i m | x | → ∞ u ( x ) = 0 , where Ω = x ∈ ℝ N : | x | > r ₀ , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and 0 < ∫ r ₀ ∞ r K ( r ) d r < ∞ , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the fixed point...

Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Jia-Feng Liao, Jiu Liu, Peng Zhang, Chun-Lei Tang (2016)

Annales Polonici Mathematici

We study the following singular elliptic equation with critical exponent ⎧ - Δ u = Q ( x ) u 2 * - 1 + λ u - γ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where Ω ⊂ ℝ N (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

Influence of diffusion on interactions between malignant gliomas and immune system

Urszula Foryś (2010)

Applicationes Mathematicae

We analyse the influence of diffusion and space distribution of cells in a simple model of interactions between an activated immune system and malignant gliomas, among which the most aggressive one is GBM Glioblastoma Multiforme. It turns out that diffusion cannot affect stability of spatially homogeneous steady states. This suggests that there are two possible outcomes-the solution is either attracted by the positive steady state or by the semitrivial one. The semitrivial steady state describes...

Liouville theorems, a priori estimates, and blow-up rates for solutions of indefinite superlinear parabolic problems

Juraj Földes (2011)

Czechoslovak Mathematical Journal

In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.

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