Tomás Caraballo, Marta Herrera-Cobos and Pedro Marín-Rubio

(\Omega)}\subset{H^{-1}(\Omega)}$ . Observe that as a result of the previous identification, we can abuse of the notation considering l ∈ L 2 (Ω) but continue denoting ( l,u ) as l ( u ). The duality product between L p (Ω) and L q (Ω) (where q is the conjugate exponent of p ) will be denoted by (·,·), and the norm in L s (Ω) will be represented by ||·|| L s (Ω) with s ≥ 1. We also denote by 〈·,·〉 the duality product between H −1 (Ω) + L q (Ω) and H 0 1 ( Ω ) ∩ L p ( Ω ) $H^1_0(\Omega)\cap{L^p(\Omega)}$ .
Definition 1
A weak solution to ( 1 ) is a