Solutions for a class of fractional boundary value problem with mixed nonlinearities
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In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem $$ \left\{ \begin{array}{ll} _tD^{\alpha}_T(_0D^{\alpha}_t u(t))=\nabla W(t,u(t)),\quad t\in [0,T],\\[0.1cm] u(0)=u(T)=0, \end{array} \right. \leqno(\mbox{FBVP}) $$ where $\alpha\in (1/2,1)$, $u\in \R^n$, $W\in C^1([0,T]\times\R^n,\R)$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. The novelty of this paper is that, when the nonlinearity $W(t,u)$ involves a combination of superquadratic and subquadratic terms, under some suitable assumptions we show that (FBVP) possesses at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved.
Abstract
In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem $$ \left\{ \begin{array}{ll} _tD^{\alpha}_T(_0D^{\alpha}_t u(t))=\nabla W(t,u(t)),\quad t\in [0,T],\\[0.1cm] u(0)=u(T)=0, \end{array} \right. \leqno(\mbox{FBVP}) $$ where $\alpha\in (1/2,1)$, $u\in \R^n$, $W\in C^1([0,T]\times\R^n,\R)$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. The novelty of this paper is that, when the nonlinearity $W(t,u)$ involves a combination of superquadratic and subquadratic terms, under some suitable assumptions we show that (FBVP) possesses at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved.
참고문헌 (21)
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