Neumann problems with indefinite and unbounded potential and concave terms

Abstract

We consider a semilinear parametric Neumann problem driven by the negative Laplacian plus an indefinite and unbounded potential. The reaction is asymptotically linear and exhibits a negative concave term near the origin. Using variational methods together with truncation and perturbation techniques and critical groups, we show that for all small values of the parameter the problem has at least five nontrivial solutions, four of which have constant sign.

Existence and multiplicity of solutions for $p$-Kirchhoff-type Neumann problems

“…The solvability of such problem has been studied by many authors. References [1], [2], [5], [6], [7], [9], [14], [15], [18], [23], [24], [25], [26], [29], [33] can be recommended to readers. In particular, when p = 2, in [18] and [30]- [31] existence and multiplicity of solutions were obtained under the Landesman-Lazer type condition and under a new Landesman-Lazer type condition, respectively.…”

(p, 2)-Equations with a Crossing Nonlinearity and Concave Terms

“…So, in the equation of [3], the concave term enters with a positive sign and in the reaction of the problem we have the competing effects of concave (sublinear) and convex (superlinear) terms. Extensions of the work of Ambrosetti-Brezis-Cerami [3] can be found in the papers of Garcia Azorero-Peral Alonso-Manfredi [7], Guo-Zhang [11], Marano-Papageorgiou [16], Papageorgiou-Rǎdulescu [22,23]. Equations in which the concave term enters with a negative sign were investigated by de Paiva-Massa [18], de Paiva-Presoto [19], Perera [26].…”

Robin problems with indefinite linear part and competition phenomena

“…Recently we have examined Robin and Neumann problems with indefinite linear part. We mention the works of Papageorgiou and Rȃdulescu [13,14,16]. In [13] the problem is parametric with competing nonlinearities.…”

“…We mention the works of Papageorgiou and Rȃdulescu [13,14,16]. In [13] the problem is parametric with competing nonlinearities. The concave term is −λ|x| q−2 x, 1 < q < 2, x ∈ R (so it enters into the equation with a negative sign) while the perturbation f (z, x) is Carathéodory, asymptotically linear near ±∞ and resonant with respect to the principal eigenvalue.…”