Resumen de Multiplicity of Positive Solutions for a Semilinear Elliptic System with Strongly Coupled Critical Terms and Concave Nonlinearities

Xiu Zhou, Hong Ying Li, Jia Feng Liao

• In this paper, the following semilinear elliptic system involving strongly coupled critical terms and concave nonlinearities is considered ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ −u = η1α1 2∗ |u| α1−2|v| β1 u + η2α2 2∗ |u| α2−2|v| β2 u + a1 |u| q−2u |x| γ , x ∈ , −v = η1β1 2∗ |u| α1 |v| β1−2v + η2β2 2∗ |u| α2 |v| β2−2v + a2 |v| q−2v |x| γ , x ∈ , u,v > 0, x ∈ , u = v = 0, x ∈ ∂, where ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary and 0 ∈ , 1 < q < 2, η1 + η2 > 0, 0 ≤ ηi < +∞, ai > 0, 0 ≤ γ < N + q − q N 2 , αi, βi > 1, αi + βi = 2∗ = 2N N−2 (i = 1, 2) is the critical Sobolev exponent. By the Nehari method and variational method, two positive solutions are obtained which generalizes and improves some corresponding results in the literature