Multiple positive solutions for the \(m\)-Laplacian and a nonlinearity with many zeros. (English) Zbl 1413.35231

Summary: In this paper, we consider the quasilinear elliptic equation \(-\Delta_mu=\lambda f(u)\), in a bounded, smooth and convex domain. When the nonnegative nonlinearity \(f\) has multiple positive zeros, we prove the existence of at least two positive solutions for each of these zeros, for \(\lambda\) large, without any hypothesis on the behavior at infinity of \(f\). We also prove a result concerning the behavior of the solutions as \(\lambda\to\infty\).


35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35B45 A priori estimates in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35D30 Weak solutions to PDEs