Existence and multiplicity of positive solutions for nonhomogeneous Dirichlet p-Laplacian problems without the Ambrosetti–Rabinowitz condition

Abstract We study the existence and multiplicity of positive weak solutions for nonhomogeneous Dirichlet problems driven by the p -Laplacian. The nonlinearity is assumed to be locally superlinear at infinity, possibly sign-changing with respect to the spatial variable, and is not required to satisfy the Ambrosetti–Rabinowitz condition. When $$p\ne 2$$ p ≠ 2 , although a nonhomogeneous boundary datum can be removed by an affine substitution, the resulting principal part is no longer p -homogeneous in the translated variable, and the classical homogeneous AR-free compactness arguments break down. To overcome this obstruction, we introduce two structural hypotheses tailored to the nonhomogeneous setting: a localized superlinearity condition and a shifted version of the function $$H(x,t)=tf(x,t)-pF(x,t)$$ H ( x , t ) = t f ( x , t ) - p F ( x , t ) . The latter compensates for the loss of admissible test functions caused by the nontrivial boundary datum. Combining the method of upper and lower solutions with Schechter’s variant of the Mountain Pass Theorem, we prove the existence of at least one positive weak solution and, under additional assumptions, the existence of a second positive weak solution. All solutions obtained are of class $$C^{1,\gamma }$$ C 1 , γ .