Multiple solutions of a parameter-dependent quasilinear elliptic equation

Consider the quasilinear elliptic equation where Ω ⊂ R N ( N ≥ 2 ) is a bounded domain with smooth boundary and λ > 0 is a parameter. For the quasilinear term, we assume that a i j = a j i and a i j ( x , s ) ’s growth is like ( 1 + s 2 ) δ i j . The nonlinearity of power growth f ( x , s ) = | s | r - 2 s with 2 < r < 4 acts as a typical example of the nonlinear term f , a case in which less results are known compared with the cases 1 < r < 2 and 4 < r < 4 N / ( N - 2 ) + . We show the structure of solutions depends keenly upon the parameter λ . More precisely, while such an equation has no nontrivial solution for λ small, we prove that both the number of solutions with positive energies and the number of solutions with negative energies tend to infinity as λ → + ∞ . Nodal properties are determined for six solutions among all of them.