Existence of nonnegative solutions for quasilinear elliptic equations with indefinite critical nonlinearities

Combined effects of concave–convex nonlinearities and indefinite potential in some elliptic problems

“…Problems with indefinite concave nonlinearities were investigated by de Paiva [6], Li, Wu and Zhou [12], and Papageorgiou and Rȃdulescu [18] only in the context of semilinear equations (that is, p = 2) and with a particular reaction of the form x → ϑ(z)x q−1 + λx r−1 for all x 0, with ϑ ∈ L ∞ (Ω) and 1 < q < 2 < r < 2 * . We also refer to the related papers by de Figueiredo, Gossez and Ubilla [5] and Narukawa and Takajo [16]. Using variational methods based on the critical point theory, combined with suitable truncation and comparison techniques, we establish the existence, nonexistence and multiplicity of positive solutions for problem (P λ ) as the parameter λ > 0 varies.…”