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-Mleft( Vert abla uVert _{L^{2}}^{2}right) Delta u+lambda V(x) u=f( x,u) & text{ in }mathbb{R}^{N}, uin H^{2}(mathbb{R}^{N}), & end{array}% right. end{equation*}% where $Ngeq 3$, $M(t) =at^{delta }+b$ with $a,delta >0$ and $bin mathbb{% R}$, $lambda >0$ is a parameter, $Vin C(mathbb{R}^{N},mathbb{R})$ and $% fin C(mathbb{R}^{N}times mathbb{R},mathbb{R}).$ 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 $% u_{a,lambda }^{(2)}rightarrow u_{infty}^{(2)}$ strongly in $H^{2}(mathbb{% R}^{N})$ as $lambdarightarrowinfty$, where $u_{infty}^{(1)} eq u_{infty}^{(2)}in H_{0}^{2}(Omega )$ are two nontrivial solutions of Dirichlet BVPs on the bounded domain $Omega$. 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.