We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u|}_{\partial \Omega }=0$, where f(x) and h(u) satisfy minimal regularity assumptions.

We give a sufficient condition for [μ-M, μ+M] × {0} to be a bifurcation interval of the equation u = L(λu + F(u)), where L is a linear symmetric operator in a Hilbert space, μ ∈ r(L) is of odd multiplicity, and F is a nonlinear operator. This abstract result provides an elementary proof of the existence of bifurcation intervals for some eigenvalue problems with nondifferentiable nonlinearities. All the results obtained may be easily transferred to the case of bifurcation from infinity.

We study the Dirichlet boundary value problem for the $p$-Laplacian of the form $$-{\Delta }_{p}u-{\lambda }_1{\left|u\right|}^{p-2}u=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $N\ge 1$, $p>1$, $f\in C\left(\overline{\Omega }\right)$ and ${\lambda }_1>0$ is the first eigenvalue of ${\Delta }_p$. We study the geometry of the energy functional $$E_p\left(u\right)=\frac{1}{p}{\int }_{\Omega }{\left|\nabla u\right|}^p-\frac{{\lambda }_1}{p}{\int }_{\Omega }{\left|u\right|}^p-{\int }_{\Omega }fu$$ and show the difference between the case $1<p<2$ and the case $p>2$. We also give the characterization of the right hand sides $f$ for which the above Dirichlet problem is solvable and has multiple solutions.

The present work is a mathematical analysis of two algorithms, namely the Roothaan and the level-shifting algorithms, commonly used in practice to solve the Hartree-Fock equations. The level-shifting algorithm is proved to be well-posed and to converge provided the shift parameter is large enough. On the contrary, cases when the Roothaan algorithm is not well defined or fails in converging are exhibited. These mathematical results are confronted to numerical experiments performed by chemists.

Let $\mathrm{\Omega }$ be a bounded open convex set of class $C^2$. Let $a\left(x,H\left(u\right)\right)$ be a non linear operator satisfying the condition (A) (elliptic) with constants $\alpha$, $\gamma$, $\delta$. We prove that a number $\lambda \ge 0$ is an eigenvalue for the operator $a\left(x,H\left(u\right)\right)$ if and only if the number $\alpha \lambda$ is an eigen-value for the operator $\mathrm{\Delta }u$. If $\lambda \ge 0$ , the two systems $a\left(x,H\left(u\right)\right)=\lambda u$ and $\mathrm{\Delta }u=\alpha \lambda u$ have the same solutions. In particular, also the eventual eigen-values of the operator $a\left(x,H\left(u\right)\right)$ should all be negative. Finally, we obtain a sufficient condition for the existence of solutions $u\in H^2\cap H_0^1\left(\mathrm{\Omega }\right)$ of the system...

We study the existence of solutions for a p-biharmonic problem with a critical Sobolev exponent and Navier boundary conditions, using variational arguments. We establish the existence of a precise interval of parameters for which our problem admits a nontrivial solution.