N-Laplacian equations in ℝ^N with subcritical and critical growth without the Ambrosetti-Rabinowitz condition

2010/12/25 by Nhuyen Lam, Guozhen Lu, Lam, Nhuyen +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1012.5489

openalex publication_date 2010/12/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let Ω be a bounded domain in ℝ^N. In this paper, we consider the following nonlinear elliptic equation of N-Laplacian type: -Δ_Nu=f(x,u) where u∈ W₀^1,2\0 when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to the above equation without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N-Laplacian in ℝ^N when the nonlinear term f has the exponential growth only deal with the case when f satisfies the (AR) condition. Our approach is based on a suitable version of the Mountain Pass Theorem introduced by G. Cerami \citeCe1, Ce2. This approach can also be used to yield an existence result for the p-Laplacian equation (1

Related

• Existence, nonexistence, and asymptotic behavior of solutions for N-Laplacian equations involving critical exponential growth in the whole ℝ^N
• Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in ℝ^N
• Multiplicity of strong solutions for a class of elliptic problems without the Ambrosetti-Rabinowitz condition in ℝ^N
• On problems with weighted elliptic operator and general growth nonlinearities
• Elliptic equations involving the p-Laplacian and a gradient term\n having natural growth