Multiplicity of positive solutions for a quasilinear Schrödinger equation with an almost critical nonlinearity

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In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem equation* { array[c]ll -Δu - Δ(u²)u = |u|ᵖ⁻²u & in Ωu= 0 & on ∂Ω, array. equation* where Ω is a smooth and bounded domain in Rⁿ,N≥3. More specifically we prove that, for p near the critical exponent 22*=4N/(N-2), the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick-Schnirelmann category and the Poincaré polynomial.