Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 41, pp. 1-23.

Title: Multiplicity and concentration of nontrivial solutions for 
       generalized extensible beam equations in R^N

Authors: Juntao Sun (Shandong Univ. of Technology, Zibo, China)
         Tsung-Fang Wu (National Univ. of Kaohsiung, Kaohsiung, Taiwan)
 
Abstract:
 In this article, we study a class of generalized extensible beam equations
 with a superlinear nonlinearity
 $$
 \Delta ^2u-M\big( \| \nabla u\| _{L^2}^2\big) \Delta u
 +\lambda V(x) u=f( x,u) \quad \text{in }\mathbb{R}^N, \quad
 u\in H^2(\mathbb{R}^N),
 $$
 where $N\geq 3$, $M(t) =at^{\delta }+b$ 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
 non-positive, 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
 blows up as the nonlocal term vanishes. Moreover,
 $u_{a,\lambda }^{(1)}\to u_{\infty}^{(1)}$ and
 $u_{a,\lambda }^{(2)}\to u_{\infty}^{(2)}$ strongly in
 $H^2(\mathbb{R}^N)$ as $\lambda\to\infty$, where
 $u_{\infty}^{(1)}\neq u_{\infty}^{(2)}\in H_0^2(\Omega )$ are two
 nontrivial solutions of Dirichlet BVPs on the bounded domain $\Omega$.
 Also, the nonexistence of  nontrivial solutions is also obtained for <em>a</em> 
 large enough.


Submitted December 1, 2018. Published March 19, 2019.
Math Subject Classifications: 35J30, 35J35.
Key Words: Extensible beam equations; nontrivial solution;  multiplicity; 
           concentration of solutions.