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

TL;DR

This paper identifies precise conditions under which an elliptic boundary value problem with oscillating nonlinearities admits infinitely many solutions or none, using variational and ODE methods for different operator classes.

Contribution

It provides new sharp criteria for existence and nonexistence of infinitely many solutions in elliptic problems with oscillating nonlinearities, covering both divergence and non-divergence form operators.

Findings

Established parameter ranges for solution existence and nonexistence.

Applied variational methods for divergence form operators.

Used ODE arguments for non-divergence form operators.

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.