Constant-sign and sign-changing solutions for nonlinear eigenvalue problems. (English) Zbl 1138.35075
Summary: We prove the existence of multiple constant-sign and sign-changing solutions for a nonlinear elliptic eigenvalue problem under Dirichlet boundary condition involving the \(p\)-Laplacian. More precisely, we establish the existence of a positive solution, of a negative solution, and of a nontrivial sign-changing solution when the eigenvalue parameter \(\lambda \) is greater than the second eigenvalue \(\lambda _{2}\) of the negative \(p\)-Laplacian, extending results by Ambrosetti-Lupo, Ambrosetti-Mancini, and Struwe. Our approach relies on a combined use of variational and topological tools (such as, e.g., critical points, mountain-pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the \(p\)-Laplacian) and comparison arguments for nonlinear differential inequalities. In particular, the existence of extremal nontrivial constant-sign solutions plays an important role in the proof of sign-changing solutions.
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 49J40 | Variational inequalities |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
Keywords:
nonlinear eigenvalue problem; \(p\)-Laplacian; critical points; mountain-pass theorem; deformation lemma; constant-sign solutions; sign-changing solutionsReferences:
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