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arXiv 1812.03043math.AP
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广义可伸长梁方程非平凡解的多重性和集中性

Multiplicity and concentration of nontrivial solutions for the generalized extensible beam equations

Juntao Sun, Tsung-fang Wu

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摘要：

本文研究了一类具有超线性非线性项\begin{equation*} \left\{ \begin{array}{ll} Δ^{2}u-M\left( \Vert \nabla u\Vert _{L^{2}}^{2}\right) Δu+λV(x) u=f( x,u) & \text{ in }\mathbb{R}^{N}, \\ u\in H^{2}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*} %的广义可伸长梁方程其中$N\geq 3$，$M(t) =at^{δ}+b$ with $a,δ>0$和$b\in \mathbb{% R}$，$λ>0$是参数，$V\in C(\mathbb{R}^{N},\mathbb{R})$和$% f\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R}).$与大多数关于这个问题的文献不同，我们允许常数$b$是非正的，这是有物理意义的.在对$V(x)$和$f(x,u)$的适当假设下，当$a$很小而$λ$足够大时，我们证明了两个非平凡解$u_{a,λ}^{(1)}$和$% u_{a,λ}^{(2)}$的存在性，其中一个解将随着非局部项的消失而爆破.此外，$u_{a,λ}^{(1)}\rightarrow u_{\infty}^{(1)}$和$% u_{a,λ}^{(2)}\rightarrow u_{\infty}^{(2)}$强地在$H^{2}(\mathbb{% R}^{N})$中为$λ\rightarrow\infty$，其中$u_{\infty}^{(1)}\neq u_{\infty}^{(2)}\in H_{0}^{2}(Ω)$是有界域$Ω$上Dirichlet边值问题的两个非平凡解.值得注意的是，这里探索了弱解$u_{\infty}^{(i)}(i=1,2)$的正则性。最后，对于足够大的$a$，也得到了非平凡解的不存在性。

英文摘要：

In this paper, we study a class of generalized extensible beam equations with a superlinear nonlinearity \begin{equation*} \left\{ \begin{array}{ll} Δ^{2}u-M\left( \Vert \nabla u\Vert _{L^{2}}^{2}\right) Δu+λV(x) u=f( x,u) & \text{ in }\mathbb{R}^{N}, \\ u\in H^{2}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 3$, $M(t) =at^{δ}+b$ with $a,δ>0$ and $b\in \mathbb{% R}$, $λ>0$ is a parameter, $V\in C(\mathbb{R}^{N},\mathbb{R})$ and $% f\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R}).$ Unlike most other papers on this problem, we allow the constant $b$ to be nonpositive, which has the physical significance. Under some suitable assumptions on $V(x)$ and $f(x,u)$, when $a$ is small and $λ$ is large enough, we prove the existence of two nontrivial solutions $u_{a,λ}^{(1)}$ and $% u_{a,λ}^{(2)}$, one of which will blow up as the nonlocal term vanishes. Moreover, $u_{a,λ}^{(1)}\rightarrow u_{\infty}^{(1)}$ and $% u_{a,λ}^{(2)}\rightarrow u_{\infty}^{(2)}$ strongly in $H^{2}(\mathbb{% R}^{N})$ as $λ\rightarrow\infty$, where $u_{\infty}^{(1)}\neq u_{\infty}^{(2)}\in H_{0}^{2}(Ω)$ are two nontrivial solutions of Dirichlet BVPs on the bounded domain $Ω$. It is worth noting that the regularity of weak solutions $u_{\infty}^{(i)}(i=1,2)$ here is explored. Finally, the nonexistence of nontrivial solutions is also obtained for $a$ large enough.

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