Solutions for a quasilinear elliptic problem with indefinite nonlinearity with critical growth

Costa, Gustavo S. A. and Figueiredo, Giovany M. and Junior, José Carlos O. (2023) Solutions for a quasilinear elliptic problem with indefinite nonlinearity with critical growth. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (24). pp. 1-19. ISSN 1417-3875

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Abstract

We are interested in nonhomogeneous problems with a nonlinearity that changes sign and may possess a critical growth as follows −div a(|∇u|p)|∇u|p−2∇u = λ|u|q−2u+W(x)|u|r−2u in Ω, u =0 on ∂Ω, where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, N ≥ 2, 1 < p ≤ q < N, q < r ≤ q∗, λ ∈ Rand function W is a weight function which changes sign in Ω. Using variational methods, we prove the existence of four solutions: two solutions which do not change sign and two solutions which change sign exactly once in Ω.

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