We investigate the behavior of weak solutions to the transmission problem for the Laplace operator with N different media in a neighborhood of a boundary conical point. We establish a precise exponent of the decreasing rate of the solution.

We present a pair of conjectural formulas that compute the leading term of the spectral asymptotics of a Schrödinger operator on $L^2\left({ℝ}^n\right)$ with quasi-homogeneous polynomial magnetic and electric fields. The construction is based on the orbit method due to Kirillov. It makes sense for any nilpotent Lie algebra and is related to the geometry of coadjoint orbits, as well as to the growth properties of certain “algebraic integrals,” studied by Nilsson. By using the direct variational method, we prove that the...

This paper is devoted to local static bifurcation theory for a class of degenerate boundary value problems for nonlinear second-order elliptic differential operators. The purpose of this paper is twofold. The first purpose is to prove that the first eigenvalue of the linearized boundary value problem is simple and its associated eigenfunction is positive. The second purpose is to discuss the changes that occur in the structure of the solutions as a parameter varies near the first eigenvalue of the...

We prove existence and bifurcation results for a semilinear eigenvalue problem in ${ℝ}^N$$(N\ge 2)$, where the linearization — $$ has no eigenvalues. In particular, we show...

We deal with a bifurcation result for the Dirichlet problem ⎧$-{\Delta }_{p}u=\mu /{\left|x\right|}^p{\left|u\right|}^{p-2}u+\lambda f(x,u)$ a.e. in Ω, ⎨ ⎩$u_{|\partial \Omega }=0$. Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $\lambda {*}_{\mu }$ such that for every $\lambda \in ]0,\lambda {*}_{\mu }[$ the above problem admits a nonzero weak solution $u_{\lambda }$ in $W{₀}^{1,p}\left(\Omega \right)$ satisfying $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }\left|\right|=0$.

This paper was inspired by the works of P. H. Rabinowitz. We study nonlinear eigenvalue problems for some fourth order elliptic partial differential equations with nonlinear perturbation of Rabinowitz type. We show the existence of an unbounded continuum of nontrivial positive solutions bifurcating from (μ₁,0), where μ₁ is the first eigenvalue of the linearization about 0 of the considered problem. We also prove the related theorem for bifurcation from infinity. The results obtained are similar...

A bifurcation problem for the equation $$\Delta u+\lambda u-\alpha u^{+}+\beta u^{-}+g(\lambda ,u)=0$$ in a bounded domain in ${}^N$ with mixed boundary conditions, given nonnegative functions $\alpha ,\beta \in L_{\infty }$ and a small perturbation $g$ is considered. The existence of a global bifurcation between two given simple eigenvalues ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}$ of the Laplacian is proved under some assumptions about the supports of the functions $\alpha ,\beta$. These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}$.

Let L = -Δ + V be a Schrödinger operator in ${ℝ}^d$ and $H{¹}_L\left({ℝ}^d\right)$ be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by $T^{\pm }(f,g)\left(x\right)=\left(T₁f\right)\left(x\right)\left(T₂g\right)\left(x\right)\pm \left(T₂f\right)\left(x\right)\left(T₁g\right)\left(x\right)$, where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T¯ is bounded from $L^p\left({ℝ}^d\right)\times L^q\left({ℝ}^d\right)$ to $H{¹}_L\left({ℝ}^d\right)$ for 1 < p,q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.