Existence of nontrivial solutions for p-Laplacian-Like equations

Abstract

In this paper, we study the p-Laplacian-Like equations involving Hardy potential or involving critical exponent and prove the existence of one or infinitely many nontrivial solutions. The results of the equations discussed can be applied to a variety of different fields in applied mechanics.

N-Laplacian Equations in ℝN with Subcritical and Critical GrowthWithout the Ambrosetti-Rabinowitz Condition

“…Over the years, many researchers studied problem (1.1) by trying to drop the (AR) condition, see for instance [17,18,20,21,22,23,24,25,31,34,35,37,39]. For example, the following assumption has been studied by many authors:…”

“…f (x, t) |t| p−1 is non-decreasing with respect to |t| (see [24,25,35] and references therein). Recently, the authors of [14] have used the following condition:…”

Existence of Nontrivial Solutions for Perturbedp-Laplacian Equation inℝNwith Critical Nonlinearity

“…For general > 1, most of the work (see [17][18][19] and the reference therein) dealt with (1) with = 1, ( ) ≡ 0 and a certain sign potential ( ). Liu and Zheng [20] considered the above mentioned problem with sign-changing potential and subcritical -superlinear nonlinearity. Cao et al [21] also studied the similar problem.…”

The existence of infinitely many solutions for p-Laplacian type equations on R^N with linking geometry

“…For λ ≥ µ 1 , there exists some m ≥ 1 such that µ m ≤ λ < µ m+1 , then we could prove that I possesses a linking structure over cones under the condition (H p ). By the linking theorem over cones under (C) c condition (see, e.g., Lemma 2.2 in [25]), we can get the existence of one nontrivial solution for problem (1.1). For 0 ≤ λ < µ 1 , we could obtain the existence result by mountain pass theorem.…”