Choquard type equations with asymptotically linear nonlinearities

Abstract

In this work, we consider the following class nonlocal elliptic problems: where , denotes the Riesz potential with , is a positive continuous potential and the nonlinearity is asymptotically linear at infinity and at the origin in a suitable sense. The nonlocal term is the convolution de with the primitive F(x, u) of f(x, u). Our main results rely on the fact that nonlocal semilinear elliptic problems have nontrivial solutions whenever a kind of crossing of eigenvalues is allowed. Here, we consider an eigenvalue elliptic problem with a nonlocal term driven by the Choquard equation.