Two different sequences of fast homoclinic solutions for damped vibration systems with $p$-Laplacian
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This article concerns the existence of infinitely many fast homoclinic solutions for the following damped vibration system with $p$-Laplacian $$\aligned &\frac{d}{dt}\Big(\left|\dot{u}(t)\right|^{p-2}\dot{u}(t)\Big)+q(t)\left|\dot{u}(t)\right|^{p-2}\dot{u}(t)\\ &-V(t)\left|u(t)\right|^{p-2}u(t)+\nabla W(t,u(t))=0\endaligned$$ where $p\geq 2$, $q,V\in C(\mathbb{R},\mathbb{R})$ and $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. Applying Fountain Theorem and Dual Fountain Theorem, we prove that above system possesses two different sequences of fast homoclinic solutions when $V$ satisfies a new coercive condition and the potential $W(t,x)$ is a combination of a superquadratic and a subquadratic functions.
Abstract
This article concerns the existence of infinitely many fast homoclinic solutions for the following damped vibration system with $p$-Laplacian $$\aligned &\frac{d}{dt}\Big(\left|\dot{u}(t)\right|^{p-2}\dot{u}(t)\Big)+q(t)\left|\dot{u}(t)\right|^{p-2}\dot{u}(t)\\ &-V(t)\left|u(t)\right|^{p-2}u(t)+\nabla W(t,u(t))=0\endaligned$$ where $p\geq 2$, $q,V\in C(\mathbb{R},\mathbb{R})$ and $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. Applying Fountain Theorem and Dual Fountain Theorem, we prove that above system possesses two different sequences of fast homoclinic solutions when $V$ satisfies a new coercive condition and the potential $W(t,x)$ is a combination of a superquadratic and a subquadratic functions.