On semilinear elliptic equations with indefinite nonlinearities

Abstract

Abstract. This paper concerns semilinear elliptic equations whose nonlinear term has the form W(x)f(u) where W changes sign. We study the existence of positive solutions and their multiplicity. The important role played by the negative part of W is contained in a condition which is shown to be necessary for homogeneous /. More general existence questions are also discussed.

A concave—convex elliptic problem involving the fractional Laplacian

“…For α ∈ [1,2) and p subcritical, we also prove that there exists a universal L ∞ -bound for every solution to problem (1.1) independently of λ. …”

“…It is essential to have that the first solution is given as a local minimum of the associated functional, J λ . To prove this last assertion we follow some ideas developed in [2].…”

On branches of positive solutions for 𝑝-Laplacian problems at the extreme value of the Nehari manifold method

“…An answer for this question, in the case p = 2, follows from the works of Alama & Tarantello [1], Ouyang [13], where the authors proved that (1.1) possess a branch of minimal positive solution for λ belonging to the whole interval (−∞, Λ) and does not admit any positive solutions for λ > Λ. However, one approach used in [1,13] is based on the application of the local continuation method [3], which essentially involves an analysis of the corresponding linearized problems.…”

“…The problems with the indefinite nonliearity of type (1.1) have been intesively studied, see e.g., Alama & Tarantello [1], Berestycki, Capuzzo-Dolcetta & Nirenberg [2], Ouyang [13]. One of the fruitful approaches in the study of such problems is the Nehari manifold method [12] where solutions are obtained through the constrained minimization problem min{Φ λ (u) : u ∈ N λ } (1.2) with the Nehari manifold N λ := {u ∈ W \0 : D u Φ λ (u)(u) = 0} (see e.g.…”

Mountain pass theorem in ordered Banach spaces and its applications to semilinear elliptic equations

“…Similar problems have been studied by Alama and Tarantello [2] and Li and Wang [12]. However the assumptions below are more general.…”