The nonlocal bistable equation: Stationary solutions on a bounded interval (Q2762922)

The following semilinear integral equation is studied on the interval (0,1): NEWLINE\[NEWLINE-J[u]+ju+ f(u)=0,NEWLINE\]NEWLINE where NEWLINE\[NEWLINEJ[u](x)= \int^1_0 J(x,y)u(y) dy, \quad j(x)= \int^1_0 J(x,y) dy.NEWLINE\]NEWLINE Here it is assumed that \(J\) is a symmetric function from \(W^{1,1}\) and \(f\in C^1\) has zeros at the points \(-1,1\) and \(a\in (-1,1)\). Moreover, \(f'(-1)>0\), \(f'(1)>0\) and \(f'(a)<0\). The equation (1) arises in some considerations of variational calculus. It is shown that if \(ju+f(u)\) is monotone then for a wide class of \(J\)'s nonconstant local minimizers do not exist. In the case when \(ju+fu\) is nonmonotone it is shown that variations along non-smooth path lead to a nonexistence result. With help of the method of iteration monotone solutions of (1) are constructed.