A priori bounds and multiplicity results for slightly superlinear and sublinear elliptic [formula omitted]-Laplacian equations

Abstract

We consider the following problem −Δpu=h(x,u)inΩ, u∈W01,p(Ω), where Ω is a bounded domain in RN, 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 p-superlinear at infinity. The function g is a p-sublinear function at zero. The coefficients a and b belong to Lk(Ω) for some k>Np and they are without sign condition. Firstly, we show an a priori bound on the nonnegative solutions, then by using variational arguments, we prove the existence of at least two nonnegative nontrivial solutions. One of the main difficulties is that the nonlinearity term h(x,u) does not satisfy the standard Ambrosetti and Rabinowitz condition.