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Nehari manifold and multiplicity results for a class of fractional boundary value problems with $p$-Laplacian

Ziheng Zhang|Abdeljabbar Ghanmi
대한수학회 (The Korean Mathematical Society)|2019.09|대한수학회보 (Bulletin of the KMS)|제56권 제5호|pp. 1297-1314|DOI 10.4134/BKMS.b181172

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boundary value problemexistence of solutionsNehari manifoldnonlinear fractional differential equations
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초록

In this work, we investigate the following fractional boundary value problems \begin{eqnarray*} \left\{\begin{array}{ll} _{t}D_{T}^{\alpha}\left(|_{0}D_{t}^{\alpha}(u(t))|^{p-2} {_0 D}_{t}^{\alpha}u(t)\right)\\ =\nabla W(t,u(t))+\lambda g(t) |u(t)|^{q-2}u(t),\;t\in (0,T),\\[0.2em] u(0)=u(T)=0, \end{array} \right. \end{eqnarray*} where $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$ and $W\in C([0,T]\times \mathbb{R}^{n},\mathbb{R})$ is homogeneous of degree $r$, $\lambda$ is a positive parameter, $g\in C([0,T])$, $1<r<p<q$ and $\frac{1}{p}<\alpha<1$. Using the Fibering map and Nehari manifold, for some positive constant $\lambda_0$ such that $0<\lambda<\lambda_0$, we prove the existence of at least two non-trivial solutions.

Abstract

In this work, we investigate the following fractional boundary value problems \begin{eqnarray*} \left\{\begin{array}{ll} _{t}D_{T}^{\alpha}\left(|_{0}D_{t}^{\alpha}(u(t))|^{p-2} {_0 D}_{t}^{\alpha}u(t)\right)\\ =\nabla W(t,u(t))+\lambda g(t) |u(t)|^{q-2}u(t),\;t\in (0,T),\\[0.2em] u(0)=u(T)=0, \end{array} \right. \end{eqnarray*} where $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$ and $W\in C([0,T]\times \mathbb{R}^{n},\mathbb{R})$ is homogeneous of degree $r$, $\lambda$ is a positive parameter, $g\in C([0,T])$, $1<r<p<q$ and $\frac{1}{p}<\alpha<1$. Using the Fibering map and Nehari manifold, for some positive constant $\lambda_0$ such that $0<\lambda<\lambda_0$, we prove the existence of at least two non-trivial solutions.

참고문헌 (33)

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