We consider the linear eigenvalue problem -Δu = λV(x)u, $u\in D_0^{1,2}\left(\Omega \right)$, and its nonlinear generalization $-{\Delta }_{p}u=\lambda V\left(x\right){\left|u\right|}^{p-2}u$, $u\in D_0^{1,p}\left(\Omega \right)$. The set Ω need not be bounded, in particular, $\Omega ={ℝ}^N$ is admitted. The weight function V may change sign and may have singular points. We show that there exists a sequence of eigenvalues ${\lambda }_n\to \infty$.

We consider the nonlinear eigenvalue problem $$-{div\left(\right|\nabla u|}^{p-2}{\nabla u)=\lambda g\left(x\right)|u|}^{p-2}u$$ in ${𝐑}^N$ with $p>1$. A condition on indefinite weight function $g$ is given so that the problem has a sequence of eigenvalues tending to infinity with decaying eigenfunctions in $W^{1,p}\left({𝐑}^N\right)$. A nonexistence result is also given for the case $p\ge N$.

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$-Laplacian and the Navier $p$-biharmonic operator on a ball of radius $R$ in ${ℝ}^N$ and its asymptotics for $p$ approaching $1$ and $\infty$. Let $p$ tend to $\infty$. There is a critical radius $R_C$ of the ball such that the principal eigenvalue goes to $\infty$ for $0<R\le R_C$ and to $0$ for $R>R_C$. The critical radius is $R_C=1$ for any $N\in \mathbb{N}$ for the $p$-Laplacian and $R_C=\sqrt{2N}$ in the case of the $p$-biharmonic operator. When $p$ approaches $1$, the principal eigenvalue of the Dirichlet...

We study eigenvalue problems with discontinuous terms. In particular we consider two problems: a nonlinear problem and a semilinear problem for elliptic equations. In order to study the existence of solutions we replace these two problems with their multivalued approximations and, for the first problem, we estabilish an existence result while for the second problem we prove the existence of multiple nontrivial solutions. The approach used is variational.

Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧${\Delta \left(\right|\Delta u|}^{p-2}{\Delta u\left)-div\right(\left|\nabla u\right|}^{p-2}\nabla u)=\lambda f(x,u)+\mu g(x,u)$ in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.

The purpose of this paper is to extend the Díaz-Saá’s inequality for the unbounded domains as RN.The proof is based on the Picone’s identity which is very useful in problems involving p-Laplacian. In a second part, we study some properties of the first eigenvalue for a system of p-Laplacian. We use Díaz-Saá’s inequality to prove uniqueness and Egorov’s theorem for the isolation. These results generalize J. Fleckinger, R. F. Manásevich, N. M. Stavrakakis and F. de Thélin’s work [9] for the first...