Quasilinear Schrödinger Equations with a Singular Operator and Critical or Supercritical Growth.

We consider the following singular quasilinear Schrödinger equations involving critical exponent - Δ u - α 2 Δ (| u | α ) | u | α - 2 u = θ | u | k - 2 u + | u | 2 ∗ - 2 u + λ f (u) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , <graphic href="40840_2024_1691_Article_Equ52.gif"></graphic> where 0 < α < 1 . By using the variational methods, we first prove that for small values of λ and θ , the above problem has infinitely many distinct solutions with negative energy. Besides, we point out that odd assumption on f is required; the problem has at least one nontrivial solution. Finally, a new modified technique is used to consider the existence of infinitely many solutions for far more general equations. [ABSTRACT FROM AUTHOR]