N-Laplacian equations inℝNwith critical growth

N−laplacian Equations in Rn with Critical Growth

We study the existence of nontrivial solutions to the following problem: { u ∈ W 1,N (R ), u ≥ 0 and −div(| ∇u |N−2 ∇u) + a (x) | u |N−2 u = f(x, u) in R (N ≥ 2), where a is a continuous function which is coercive, i.e., a (x) → ∞ as | x |→ ∞ and the nonlinearity f behaves like exp ( α | u |N/(N−1) ) when | u |→ ∞.

2 01 0 N - Laplacian equations in R N with subcritical and critical growth without the Ambrosetti - Rabinowitz condition

Let Ω be a bounded domain in R . In this paper, we consider the following nonlinear elliptic equation of N -Laplacian type: (0.1) { −∆Nu = f (x, u) u ∈ W 1,2 0 (Ω) \ {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 (0.1) without the Ambrosett...

Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in RN

In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form −div(a(x,∇u))+ V (x)|u|N−2u= f (x,u) |x|β + εh(x) (0.1) inRN when f :RN ×R→R behaves like exp(α|u|N/(N−1))when |u| →∞ and satisfies the Ambrosetti– Rabinowitz condition. In particular, in the case of N -Laplacian, i.e., a(x,∇u) = |∇u|N−2∇u, we obtain multiplicity of weak solutions of (0.1). ...

Positive solutions for asymptotically periodic Kirchhoff-type equations with critical growth

In this paper‎, ‎we consider the following Kirchhoff-type equations‎: ‎$-‎left(a+bint_{mathbb{R}^{3}}|nabla u|^{2}right)Delta u+V(x) u=lambda$ $f(x,u)+u^{5}‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$u(x)>0‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$uin H^{1}(mathbb{R}^{3})‎ ,‎$ ‎ ‎‎‎where $a,b>0$ are constants and $lambda$ is a positive parameter‎. ‎The aim of this paper is to study the existence of positive ...

The Laplacian, Second Order Elliptic Equations