Existence and nonexistence of infinitely many solutions to elliptic problems with oscillating nonlinearities

Abstract

We study sharp conditions for the existence and nonexistence of infinitely many nonnegative solutions to the problem $-\Delta_p u = \lambda f(u)$ in a bounded domain with Dirichlet boundary conditions, where $f$ is a continuous function with a sequence of positive zeros converging to zero or diverging to infinity. Under a growth condition on the primitive $F(s) = \int_0^s f(t)dt$, we establish ranges of the parameter $\lambda$ for which infinitely many small or large solutions exist, as well as ranges where no bifurcation from zero or infinity can occur. The existence result is obtained via variational methods for a general class of divergence form operators, while the nonexistence result is established both for the $p$-Laplacian and for uniformly elliptic operators in non-divergence form via an ODE argument.