Given that $\left(M,g\right)$ is a smooth compact and symmetric Riemannian $n$-manifold, $n\ge 2$, we prove a multiplicity result for antisymmetric sign changing solutions of the problem $-{\epsilon }^2{\Delta }_{g}u+u={\left|u\right|}^{p-2}u$ in $M$. Here $p>2$ if $n=2$ and if $n\ge 3$.

The asymptotic behaviour of the second eigenvalue of the $p$-Laplacian operator as $p$ goes to 1 is investigated. The limit setting depends only on the geometry of the domain. In the particular case of a planar disc, it is possible to show that the second eigenfunctions are nonradial if $p$ is close enough to 1.

The study of multiple solutions for quasilinear elliptic problems under Dirichlet or nonlinear Neumann type boundary conditions has received much attention over the last decades. The main goal of this paper is to present multiple solutions results for elliptic inclusions of Clarke's gradient type under Dirichlet boundary condition involving the $p$-Laplacian which, in general, depend on two parameters. Assuming different structure and smoothness assumptions on the nonlinearities generating the multivalued term, we prove the existence of multiple constant-sign and sign-changing (nodal) solutions for parameters specified in terms of the Fučik spectrum of the $p$-Laplacian. Our approach will be based on truncation techniques and comparison principles (sub-supersolution method) for elliptic inclusions combined with variational and topological arguments for, in general, nonsmooth functionals, such as, critical point theory, Mountain Pass Theorem, Second Deformation Lemma, and the variational characterization of the “beginning”of the Fučik spectrum of the $p$-Laplacian. In particular, the existence of extremal constant-sign solutions and their variational characterization as global (resp., local) minima of the associated energy functional will play a key-role in the proof of sign-changing solutions.

Assume that $Q$ is a positive continuous function in ${\mathbb{ℝ}}^N$ and satisfies some suitable conditions. We prove that the quasilinear elliptic equation $-{\Delta }_{p}u+|u{|}^{p-2}u=Q(z\left)\right|u{|}^{q-2}u$ in ${\mathbb{ℝ}}^N$ admits at least two solutions in ${\mathbb{ℝ}}^N$ (one is a positive ground-state solution and the other is a sign-changing solution).

By combining the embedding arguments and the variational methods, we obtain infinitely many solutions for a class of superlinear elliptic problems with the Robin boundary value under weaker conditions.

We study the existence and asymptotic behavior of positive solutions for a class of quasilinear elliptic systems in a smooth boundary via the upper and lower solutions and the localization method. The main results of the present paper are new and extend some previous results in the literature.