Double phase equations with an indefinite concave term

Abstract

We consider a Dirichlet problem having a double phase differential operator with unbalanced growth and reaction involving the combined effects of a concave (sublinear) and of a convex (superlinear) terms. We allow the coefficient \(\mathcal E\in L^\infty(\Omega)\) of the concave term to be sign changing. We show that when \(\|\mathcal E\|_\infty \) is small the problem has at least two bounded positive solutions.