A priori bounds and multiplicity results for slightly superlinear and sublinear elliptic p-Laplacian equations

TL;DR

This paper establishes a priori bounds and multiplicity results for solutions to a class of p-Laplacian elliptic equations with nonlinearities that are superlinear at infinity and sublinear at zero, without sign restrictions on coefficients.

Contribution

It introduces new a priori bounds and proves the existence of multiple solutions for equations with nonlinearities lacking the Ambrosetti-Rabinowitz condition.

Findings

Established a priori bounds for solutions.

Proved existence of at least two nonnegative solutions.

Handled nonlinearities without standard growth conditions.

Abstract

We consider the following problem $ -\Delta_{p}u= h(x,u) \mbox{ in }\Omega$, $u\in W^{1,p}_{0}(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $1<p<N$, with a smooth boundary. In this paper we assume that $h(x,u)=a(x)f(u)+b(x)g(u)$ such that $f$ is regularly varying of index $p-1$ and superlinear at infinity. The function $g$ is a $p$-sublinear function at zero. The coefficients $a$ and $b$ belong to $L^{k}(\Omega)$ for some $k>\frac{N}{p}$ and they are without sign condition. Firstly, we show a priori bound on solutions, then by using variational arguments, we prove the existence of at least two nonnegative solutions. One of the main difficulties is that the nonlinearity term $h(x,u)$ does not satisfy the standard Ambrosetti and Rabinowitz condition.