Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity
Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity
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DOI：
10.1515/anona-2020-0168
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发表时间：
2021-01
影响因子：
4.2
通讯作者：
Qihan He;Juntao Lv;Zongyan Lv
中科院分区：
文献类型：
--
作者：
Qihan He;Juntao Lv;Zongyan Lv
摘要 我们考虑以下 p 调和问题 Δ(|Δu|p−2Δu)+m|u|p−2u=f(x,u),xεRN,uεW2,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 是一个常数，N > 2p ≥ 4 且 limt→∞f(x,t)|t|p−2t=l $\begin{array}{} \displaystyle \lim\limits_{t\rightarrow \infty}\frac{f(x,t)}{|t|^{p-2}t}=l \end{array}$ 在 x 中一致，这意味着 f(x, t)不满足 Ambrosetti-Rabinowitz 类型条件。通过显示上述 p 调和方程有限问题弱解的 Pohozaev 恒等式并使用 Mountain Pass 定理的变体版本，我们证明了上述方程非平凡解的存在性和不存在性。此外，如果f(x, u) ≡ f(u)，则利用人工约束方法和Pohozaev恒等式也证明了上述问题的基态解的存在性和非平凡解的不存在性。
Abstract We consider the following p-harmonic problem Δ(|Δu|p−2Δu)+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 > 2p ≥ 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.