Existence and uniqueness of solution for a singular elliptic differential equation

Abstract

Abstract In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: − Δ u − 1 2 ( x ⋅ ∇ u ) = μ h ( x ) u q − 1 + λ u − u p , x ∈ R N , u ( x ) → 0 , as ∣ x ∣ → + ∞ , \left\{\begin{array}{l}-\Delta u-\frac{1}{2}\left(x\cdot \nabla u)=\mu h\left(x){u}^{q-1}+\lambda u-{u}^{p},\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| x| \to +\infty ,\end{array}\right. where N ⩾ 3 N\geqslant 3 , 0 < q < 1 0\lt q\lt 1 , λ > 0 \lambda \gt 0 , p > 1 p\gt 1 , μ > 0 \mu \gt 0 is a parameter and the function h ( x ) h\left(x) satisfies certain conditions. To start with, based on the variational argument and perturbation method, we obtain the existence and uniqueness of the positive solution for the aforementioned singular elliptic differential equation as λ > N 2 \lambda \gt \frac{N}{2} . In addition, there is no solution as λ ⩽ N 2 \lambda \leqslant \frac{N}{2} . Later, from an experimental point of view, we give the numerical solution of the aforementioned singular elliptic differential equation by means of a neural network in some special cases, which enrich the theoretical results. Our conclusions partially extend the results corresponding to the nonsingular case.