Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity
| Autor: | Zongyan Lv, Qihan He, Juntao Lv |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
QA299.6-433
media_common.quotation_subject 010102 general mathematics Mathematical analysis Harmonic (mathematics) Infinity 01 natural sciences 35j60 010101 applied mathematics nontrivial solutions Nonlinear system p-harmonic equation 35j35 0101 mathematics ambrosetti-rabinowitz condition 47j30 Analysis Mathematics media_common |
| Zdroj: | Advances in Nonlinear Analysis, Vol 10, Iss 1, Pp 1039-1060 (2021) |
| ISSN: | 2191-9496 |
| Popis: | We consider the following p-harmonic problem Δ ( | Δ u | p − 2 Δ u ) + m | u | p − 2 u = f ( x , u ) , x ∈ R N , u ∈ W 2 , p ( R N ) , $$\begin{array}{} \displaystyle \left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\ u \in W^{2,p}({\mathbb R}^N), \end{array} \right. \end{array}$$ where m > 0 is a constant, N > 2p ≥ 4 and lim t → ∞ f ( x , t ) | t | p − 2 t = 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. |
| Databáze: | OpenAIRE |
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