Multiple solutions for weighted nonlinear elliptic system involving critical exponents

Abstract

In this paper, by using the idea of category, we investigate how the shape of the graph of h(x) affects the number of positive solutions to the following weighted nonlinear elliptic system:

$\left\{ \begin{gathered} - div(|x|^{ - 2a} \nabla u)\mu \frac{u} {{|x|^{2(a + 1)} }} = \frac{\alpha } {{\alpha + \beta }}h(x)\frac{{|u|^{\alpha - 2} |v|^\beta u}} {{|x|^{b2*(a,b)} }} + \lambda K_1 (x)|u|^{q - 2} u,\operatorname{in} \Omega , \hfill \\ - div(|x|^{ - 2a} \nabla v)\mu \frac{u} {{|x|^{2(a + 1)} }} = \frac{\beta } {{\alpha + \beta }}h(x)\frac{{|u|^\alpha |v|^{\beta - 2} v}} {{|x|^{b2*(a,b)} }} + \sigma K_2 (x)|v|^{q - 2} v,\operatorname{in} \Omega , \hfill \\ u = v = 0, \hfill \\ \end{gathered} \right. $

where 0 ∈ Ω is a smooth bounded domain in ℝ^N (N ⩾ 3), λ, σ > 0 are parameters, $0 \leqslant \mu < \bar \mu _a \triangleq (\frac{{N - 2 - 2a}} {2})2$; h(x), K ₁(x) and K ₂(x) are positive continuous functions in $\bar \Omega$, 1 ⩽ q < 2, α, β > 1 and α + β = 2*(a, b) $(2*(a,b) \triangleq \frac{{2N}} {{N - 2(1 + a - b)}})$, is critical Sobolev-Hardy exponent). We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters (λ, σ) belongs to a certain subset of ℝ².