We consider a semilinear elliptic eigenvalues problem on a ball of ${\Re }^n$ and show that all the eigenfunctions and eigenvalues, can be obtained from the Lane-Emden function.

We prove the existence of the least positive eigenvalue with a corresponding nonnegative eigenfunction of the quasilinear eigenvalue problem $$\begin{array}{ccc}\hfill -div\left(a(x,u)\right|{|}^{p-2}\nabla u)=& {\lambda b(x,u)\left|u\right|}^{p-2}u\text{in}\Omega ,u=\hfill & \hfill 0\text{on}\partial \Omega ,\end{array}$$ where $\Omega$ is a bounded domain, $p>1$ is a real number and $a(x,u)$, $b(x,u)$ satisfy appropriate growth conditions. Moreover, the coefficient $a(x,u)$ contains a degeneration or a singularity. We work in a suitable weighted Sobolev space and prove the boundedness of the eigenfunction in $L^{\infty }\left(\Omega \right)$. The main tool is the investigation of the associated homogeneous eigenvalue problem and an application...

Let Ω be a bounded domain in ${ℝ}^n$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^2\left(\Omega \right)$ into itself with compact inverse, with eigenvalues $-{\lambda }_i$, each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem $Lu+b{u}^{+}-a{u}^{-}=f\left(x\right)$ in Ω, u=0 on ∂ Ω. We assume that $a<{\lambda }_1$, ${\lambda }_2<b<{\lambda }_3$ and f is generated by ${\varphi }_1$ and ${\varphi }_2$. We show a relation between the multiplicity of solutions and source terms in the equation....

We consider the $p$-Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite $p$ and investigate the limit problem as $p\to \infty$.

We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.

We prove the existence of a principal eigenvalue associated to the ∞-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.

We consider the pseudo-$p$-laplacian, an anisotropic version of the $p$-laplacian operator for $p\ne 2$. We study relevant properties of its first eigenfunction for finite $p$ and the limit problem as $p\to \infty$.

We consider the pseudo-p-Laplacian, an anisotropic version of the p-Laplacian operator for $p\ne 2$. We study relevant properties of its first eigenfunction for finite p and the limit problem as p → ∞.

We show that the spectrum of $\Delta {²}_{p}u+{2\beta ·\nabla \left(\right|\Delta u|}^{p-2}{\Delta u)+|\beta \left|²\right|\Delta u|}^{p-2}\Delta u=\alpha m{\left|u\right|}^{p-2}u$, where $\beta \in {ℝ}^N$, under Navier boundary conditions, contains at least one sequence of eigensurfaces.

General mathematical theories usually originate from the investigation of particular problems and notions which could not be handled by available tools and methods. The Fučík spectrum and the $p$-Laplacian are typical examples in the field of nonlinear analysis. The systematic study of these notions during the last four decades led to several interesting and surprising results and revealed deep relationship between the linear and the nonlinear structures. This paper does not provide a complete survey....