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Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity
Advances in Nonlinear Analysis ( IF 4.2 ) Pub Date : 2021-01-01 , DOI: 10.1515/anona-2020-0168
Qihan He 1 , Juntao Lv 1 , Zongyan Lv 1
Affiliation  

We consider the following p -harmonic problem Δ (|Δ u|p− 2Δ u)+m|u|p− 2u=f(x,u),x∈ RN,u∈ W2,p(RN), $$\begin{array}{} \displaystyle \left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\ u \in W^{2,p}({\mathbb R}^N), \end{array} \right. \end{array}$$ where m > 0 is a constant, N > 2 p ≥ 4 and limt→ ∞ f(x,t)|t|p− 2t=l $\begin{array}{} \displaystyle \lim\limits_{t\rightarrow \infty}\frac{f(x,t)}{|t|^{p-2}t}=l \end{array}$ uniformly in x , which implies that f ( x , t ) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if f ( x , u ) ≡ f ( u ), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.

中文翻译:

非线性渐近到| t | p–2t的p调和方程的非平凡解

我们考虑以下p谐波问题Δ(|Δu | p−2Δ u)+ m | u | p− 2u = f(x,u),x∈RN,u∈W2,p(RN),$$ \ begin {array} {} \ displaystyle \ left \ {\ displaystyle \ begin {array} {ll} \ displaystyle {\ it \ Delta}(| {\ it \ Delta} u | ^ {p-2} {\ it \ Delta} u)+ m | u | ^ {p-2} u = f(x,u),\ \ x \ in {\ mathbb R} ^ N,\\ u \ in W ^ {2,p} ({\ mathbb R} ^ N),\ end {array} \ right。\ end {array} $$,其中m> 0为常数,N> 2 p≥4且limt→∞f(x,t)| t | p− 2t = l $ \ begin {array} {} \ displaystyle \ lim \ limits_ {t \ rightarrow \ infty} \ frac {f(x,t)} {| t | ^ {p-2} t} = l \ end {array} $统一用x表示,这意味着f(x ,t)不满足Ambrosetti-Rabinowitz类型条件。通过显示关于上述p调和方程的有限问题的弱解的Pohozaev身份,并使用Mountain Pass定理的变体,我们证明了上述方程非平凡解的存在和不存在。此外,如果f(x,u)≡f(u),则还可以通过人工约束方法和Pohozaev身份证明存在基态解和不存在非平凡解。
更新日期:2021-01-01
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