Sign-changing solutions and multiplicity results for elliptic problems via lower and upper solutions

In the first part of this work, we recall variational methods related to invariant sets in ${C^1_0}$ . In the second part of the work, we consider an elliptic Dirichlet problem in a situation where the origin is a solution around which the nonlinearity has a slope between two consecutive eigenvalues of order larger than 2 and near + infinity the slope of the nonlinearity is smaller than the first eigenvalue. Then we discuss the conditions needed near - infinity in order to ensure the existence of a positive solution and two sign-changing solutions.