Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains

Let Ω ⊂ R be a ball or an annulus and f : R → R absolutely continuous, superlinear, subcritical, and such that f(0) = 0. We prove that the least energy nodal solution of −∆u = f(u), u ∈ H 0 (Ω), is not radial. We also prove that Fučik eigenfunctions, i. e. solutions u ∈ H 0 (Ω) of −∆u = λu − μu−, with eigenvalue (λ, μ) on the first nontrivial curve of the Fučik spectrum, are not radial. A relat...

Multiplicity of Positive and Nodal Solutions for Nonhomogeneous Elliptic Problems in Unbounded Cylinder Domains

Extrapolation of Numerical Solutions for Elliptic Problems on Corner Domains

We study the extrapolation method for the numerical solution of elliptic boundary value problems on corner domains. Most papers on the extrap-olation method for solving diierential equations require smoothness assumptions on exact solutions, which do not hold for boundary value problems on corner domains. We conjecture that asymptotic error expansions exist as long as there is enough smoothness...

Very Weak Solutions with Boundary Singularities for Semilinear Elliptic Dirichlet Problems in Domains with Conical Corners

Let Ω ⊂ R be a bounded Lipschitz domain with a cone-like corner at 0 ∈ ∂Ω. We prove existence of at least two positive unbounded very weak solutions of the problem −∆u = u in Ω, u = 0 on ∂Ω, which have a singularity at 0, for any p slightly bigger that the generalized Brezis-Turner exponent p∗. On an example of a planar polygonal domain the actual size of the p-interval on which the existence r...

Radial Symmetry of Positive Solutions of Nonlinear Elliptic Equations

We study the radial symmetry and asymptotic behavior at x = 1 of positive solutions of u = '(jxj)u ; x 2 IR n and jxj large () where > 1 and '(r) is positive and continuous for r large. In particular we give conditions on ' which guarantee that each positive solution of () satisses u(x) u(jxj) ! 1 as jxj ! 1 where u(r) is the average of u on the sphere jxj = r.