A unified approach for multiple constant sign and nodal solutions. (English) Zbl 1167.35372

Summary: We consider a nonlinear elliptic equation driven by the \(p\)-Laplacian with Dirichlet boundary condition. Using variational techniques, combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least six nontrivial solutions: two positive, two negative and two nodal (sign-changing) solutions. Our framework of analysis incorporates both coercive and \(p-1\)-superlinear problems. Also, the results on multiple constant sign solution incorporates the case of concave-convex nonlinearities.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35J70 Degenerate elliptic equations