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

We consider the following problem $-{\mathrm{\Delta }}_{p}u=h(x,u)in\mathrm{\Omega }$, $u\in W_0^{1,p}(\mathrm{\Omega })$, where $\mathrm{\Omega }$ 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 $L^k(\mathrm{\Omega })$ 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.