Asymmetric elliptic problems in .
Let be a non-negative function of class from to , which vanishes exactly at two points and . Let be the set of functions of a real variable which tend to at and to at and whose one dimensional energyis finite. Assume that there exist two isolated minimizers and of the energy over . Under a mild coercivity condition on the potential and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at and , it is possible to prove...
Let W be a non-negative function of class C3 from to , which vanishes exactly at two points a and b. Let S1(a, b) be the set of functions of a real variable which tend to a at -∞ and to b at +∞ and whose one dimensional energy is finite. Assume that there exist two isolated minimizers z+ and z- of the energy E1 over S1(a, b). Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z+ and...
Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis–Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order −1.
We give the precise behaviour of some solutions of a nonlinear elliptic B.V.P. in a bounded domain when a parameter approaches an eigenvalue of the principal part. If the nonlinearity has some regularity and the domain is for example convex, we also prove a nonlinear version of Courant’s Nodal theorem.
We give the precise behaviour of some solutions of a nonlinear elliptic B.V.P. in a bounded domain when a parameter approaches an eigenvalue of the principal part. If the nonlinearity has some regularity and the domain is for example convex, we also prove a nonlinear version of Courant's Nodal theorem.
In this paper we consider the -Laplacian problem with Dirichlet boundary condition, The term is a real odd and increasing homeomorphism, is a nonnegative function in and is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both a minimizer...
We study the leading order behaviour of positive solutions of the equation , where , and when is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of , and . The behavior of solutions depends sensitively on whether is less, equal or bigger than the critical Sobolev exponent . For the solution asymptotically coincides with the solution of the equation in which the last term is absent. For the solution asymptotically coincides...
In the recent years, many results have been established on positive solutions for boundary value problems of the form in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).