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

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 a large enough.

Keywords

• Extensible beam equations
• nontrivial solution
• multiplicity
• concentration of solutions