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

Abstract

We consider the following problem $ -Δ_{p}u= h(x,u) \mbox{ in }Ω$, $u\in W_0^{1,p}(\Omega )$, 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.

24 pages