Existence of solutions to Δ_λ-Laplace equations without the Ambrosetti–Rabinowitz condition

Abstract

We study the existence of nontrivial weak solutions for the following boundary value problemwhere is a bounded domain in R N (N ≥ 2), λ is a strongly degenerate elliptic operator, the nonlinearity f (x, u) is subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.

Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

“…Very recently, the authors of [11] investigated the ∆ γ −Laplace operator under the additional assumption that the operator is homogeneous of degree two with respect to a semigroup of dilations in R N . Many aspects of the theory of degenerate elliptic differential operators are presented in monographs [27,28] (see also some recent results in [2,3,11,12,13,14,15,16,17,18,19,22,24,26]). In this paper, we study multiplicity of weak solutions to the following problem…”

A Berestycki–Lions type result for a class of problems involving the 1-Laplacian operator

“…which has been extensively studied. See, for instance Anh and My [5], Kogoj and Lanconelli [22], Luyen and tri [28], Rahal and Hamdani [33] and references therein. At the last years some authors have dedicated a special attention for problems involving the Grushin operator, which we quote, Chung [14], D'Ambrosio [15], Monti [29], Monti and Morbidelli [30], and more recently Duong and Nguyen [16], Duong and Phan [17], Liu, Tang and Wang [26], Loiudice [27].…”

“…Related to a more general class of degenerate operators, in the year 2017, Chen, Tang and Gao studied in [13] (see also [5], [22], [28] and [33]) the following problem…”

Multiple solutions for Grushin operator without odd nonlinearity

“…Because of this reason some attempts were made to replace condition (AR). For example, in [25], Zhang-Xu also studied this problem without the (AR) condition, a, b are positive constants, V(x) ≡ 1 and f (x, u) = |u| p−2 u with p ∈ (1, 5), 2…”

“…In [5], Kogoj and Lanconelli investigated the ∆ λ -Laplace operator under the additional assumption that the operator is homogeneous of degree two with respect to a semigroup of dilations in R N . In 2015, Anh and My studied in [2] the following problem…”