The stationary Gierer–Meinhardt system in the upper half-space: existence, nonexistence and asymptotics

Abstract

AbstractWe study the existence, nonexistence and asymptotics of positive solutions to the Gierer–Meinhardt system $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda u=\frac{u^p}{v^q}+\rho (x)\,, u>0 &{} \text{ in } {\mathbb {R}}^N_+:={\mathbb {R}}^{N-1}\times (0, \infty ), \\ \displaystyle -\Delta v+\lambda v=\frac{u^m}{v^s} \,, v>0 &{} \text{ in } {\mathbb {R}}^N_+, \end{array}\right. } \end{aligned}$$ - Δ u + λ u = u p v q + ρ ( x ) , u > 0 in R + N : = R N - 1 × ( 0 , ∞ ) , - Δ v + λ v = u m v s , v > 0 in R + N , where $$N\ge 3$$ N ≥ 3 , $$\lambda >0$$ λ > 0 and $$\rho \in C({\overline{{\mathbb {R}^N_+}}})$$ ρ ∈ C ( R + N ¯ ) , $$\rho >0$$ ρ > 0 . The solutions (u, v) are assumed to satisfy the homogeneous Neumann boundary condition $$\displaystyle \frac{\partial u}{\partial x_N}=\frac{\partial v}{\partial x_N}=0$$ ∂ u ∂ x N = ∂ v ∂ x N = 0 on $$\partial {\mathbb {R}}^N_+$$ ∂ R + N and $$u(x)\rightarrow 0$$ u ( x ) → 0 , $$v(x)\rightarrow 0$$ v ( x ) → 0 as $$x\in {\mathbb {R}}^N_+$$ x ∈ R + N , $$|x|\rightarrow \infty $$ | x | → ∞ . Under various conditions on the exponents $$m,p,q,s>0$$ m , p , q , s > 0 and the data $$\rho (x)$$ ρ ( x ) we obtain qualitative properties of the solutions (u, v). In particular, we derive the existence of a solution (u, v) with u(x) having the minimal asymptotic behaviour $$u(x)\simeq \Phi _\lambda (x)$$ u ( x ) ≃ Φ λ ( x ) as $$|x|\rightarrow \infty $$ | x | → ∞ , where $$\Phi _\lambda $$ Φ λ denotes the fundamental solution of $$-\Delta +\lambda I$$ - Δ + λ I in $${\mathbb {R}^N_+}$$ R + N . The approach combines integral representations of solutions and various integral estimates with fixed point arguments.