We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger equation

\[ -\Delta v +q v =|v|^{p-2} v, \qquad \text{in}\,{ {\mathbb{R}^{3}},} \]

$2 < p < \infty$

$q \ge 0$

We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(|h|nhxxx)x, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Ltβ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.