Third-order boundary value problems with sign-changing solutions

We consider the third-order nonlinear differential equation u‴(t)=f(t,u(t),u′(t),u″(t)),a.e. t∈(0,1), satisfying u(0)=u′(0)=u″(1)=0u(0)=u′(0)=u″(1)=0 and u(0)=u′(1)=u″(1)=0,u(0)=u′(1)=u″(1)=0, where f:[0,1]×R3→Rf:[0,1]×R3→R is LpLp-Carathéodory, 1≤p<∞1≤p<∞. We obtain the existence of at least one positive solution using the Leray–Schauder Continuation Principle for each set of boundary conditions by separately considering the cases p>1p>1 and p=1p=1.