Infinitely many homoclinic solutions for different classes of fourth-order differential equations
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In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation $$u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in\mathbb{R} \leqno(1)$$ where $a(x)$ is not required to be either positive or coercive, and $F(x,u)=\int^{u}_{0}f(x,v)dv$ is of subquadratic or superquadratic growth as $\left|u\right|\rightarrow\infty$, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as $|u|\rightarrow\infty$). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.
Abstract
In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation $$u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in\mathbb{R} \leqno(1)$$ where $a(x)$ is not required to be either positive or coercive, and $F(x,u)=\int^{u}_{0}f(x,v)dv$ is of subquadratic or superquadratic growth as $\left|u\right|\rightarrow\infty$, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as $|u|\rightarrow\infty$). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.
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