Anisotropic equations with indefinite potential and competing nonlinearities

TL;DR

This paper investigates a nonlinear Dirichlet problem involving a variable exponent p-Laplacian with an indefinite potential, analyzing how the set of positive solutions changes with a parameter and establishing the existence of minimal solutions.

Contribution

It introduces a bifurcation theorem for anisotropic concave-convex problems with variable exponents and indefinite potentials, advancing understanding of solution structure.

Findings

Bifurcation phenomena characterized as the parameter varies.

Existence of minimal positive solutions established.

Analysis of solution set changes with respect to the parameter.

Abstract

We consider a nonlinear Dirichlet problem driven by a variable exponent $p$-Laplacian plus an indefinite potential term. The reaction has the competing effects of a parametric concave (sublinear) term and of a convex (superlinear) perturbation (an anisotropic concave-convex problem). We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $\lambda$ varies. Also, we prove the existence of minimal positive solutions.