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

In this paper, we study a class of generalized extensible beam equations with a superlinear nonlinearity \begin{equation*} \left{ \begin{array}{ll} \Delta ^{2}u-M\left( \Vert \nabla u\Vert {L^{2}}^{2}\right) \Delta u+\lambda V(x) u=f( x,u) & $\text{ in }$$\mathbb{R}$^{N}, \ u\in H^{2}($\mathbb{R}$^{N}), & \end{array}% \right. \end{equation*}% where $N\ge 3$, $M(t)=a{t}^{\delta }+b$ with $a,\delta >0$ and b\in \mathbb{% R}, $\lambda >0$ 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 $\lambda$ is large enough, we prove the existence of two nontrivial solutions $u{a,\lambda }^{(1)}$and$% u_{a,\lambda }^{(2)}$, one of which will blow up as the nonlocal term vanishes. Moreover, $u_{a,\lambda }^{(1)}\rightarrow u_{\infty}^{(1)}$and$% u_{a,\lambda }^{(2)}\rightarrow u_{\infty}^{(2)}$stronglyin$H^{2}(\mathbb{% R}^{N})$as$\lambda\rightarrow\infty$,where$u_{\infty}^{(1)}\neq u_{\infty}^{(2)}\in H_{0}^{2}(\Omega )$ are two nontrivial solutions of Dirichlet BVPs on the bounded domain $\mathrm{\Omega }$. It is worth noting that the regularity of weak solutions $u_{\mathrm{\infty }}^{(i)}(i=1,2)$ here is explored. Finally, the nonexistence of nontrivial solutions is also obtained for $a$ large enough.