Combined Effects of Concave and Convex Nonlinearities in Some Elliptic Problems

Three nontrivial solutions for nonlinear fractional Laplacian equations

“…x ∈ Ω (α, β > 0), and set for all (x, t) ∈ Ω × R f (x, t) = −a(x)t + b(x)|t| p−2 t. Then, f satisfies hypotheses H 2 with convenient a 0 , c 0 , and σ. This choice of f belongs in the class of concave-convex nonlinearities, whose study (in the classical case s = 1) started with [1].…”

“…Now we prove that ϕ + is unbounded from below. Indeed, letû 1 be defined as in Proposition 2.8 (i), and recall that û 1 …”

Multiple solutions to a nonlinear elliptic equation involving Paneitz type operators

“…The nonlinearity considered here is concave-convex. The simultaneous effect of the concave and convex terms has been initially investigated by Ambrosetti et al [1] in the Euclidian case for the Laplace operator. Since then, elliptic problems with this kind of nonlinearities were extensively studied by several authors with different classes of domains and with more general differential operators like the p-Laplacian.…”

Nonnegative solutions for the degenerate logistic indefinite sublinear equation

“…When 1 < m < 2 (q < 1 < p) and a(x) ≡ a 0 with a 0 a positive constant, (1.4) was studied in [4] in the particular case L = −∆ and in [6] when L is a quasilinear operator. When a changes sign, (1.4) was analyzed in [24] in the particular case λ ≤ 0.…”