933 research outputs found
Blow up and symmetry of sign changing solutions to some critical elliptic equations
No full textAbstractIn this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball
Bifurcation and asymptotic analysis for a class of supercritical elliptic problems in an exterior domain
No full textWe consider the problem {-δ u = up u > 0 u = 0 either in an annulus AR or in the domain ω = RN B̄, N ≥ 3, B = {x ε RN , |x| 2N/ N-2 . We prove that the unique radial solution up,R in AR converges, as R → +∞, to the unique fast-decay radial solution in and showseveral related asymptotic estimates, in particular spectral convergence. Analogous asymptotic estimates are also proved for nonradial uniformly bounded solutions in AR. From this we deduce that bifurcation of nonradial solutions occurs at the fast-decay degenerate radial solutions of the problem in - and that the bifurcation branches are limits, in a suitable sense, of the bifurcation branches already found in (Gladiali et al 2011 Calc. Var. Partial Diff. Eqns 40 295317). © 2011 IOP Publishing Ltd & London Mathematical Society
Symmetry results for cooperative elliptic systems in unbounded domains
No full textIn this paper we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in , or in the exterior of a ball.
We consider the case of fully coupled systems and nonlinearities which are either convex or have a convex derivative.\\
The solutions are shown to be foliated Schwarz symmetric if a bound on their Morse index holds.
As a consequence of the symmetry results we also obtain some nonexistence theorems
Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities
No full textAbstractIn this paper, we study the symmetry properties of the solutions of the semilinear elliptic problem {−Δu=f(x,u)in Ωu=g(x)on ∂Ω, where Ω is a bounded symmetric domain in RN, N⩾2, and f:Ω×R→R is a continuous function of class C1 in the second variable, g is continuous and f and g are somehow symmetric in x. Our main result is to show that all solutions of the above problem of index one are axially symmetric when Ω is an annulus or a ball, g≡0 and f is strictly convex in the second variable. To do this, we prove that the nonnegativity of the first eigenvalue of the linearized operator in the caps determined by the symmetry of Ω is a sufficient condition for the symmetry of the solution, when f is a convex function
Morse index and symmetry for elliptic problems with nonlinear mixed boundary conditions
No full textWe consider an elliptic problem of the type where Ω is a bounded Lipschitz domain in R N with a cylindrical symmetry, ν stands for the outer normal and. Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem. As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem For this one, we construct sequences of eigenvalues and provide variational characterization of them, following the usual approach for the Dirichlet case, but working in the product Hilbert space L 2 (Ω) × L 2 (Γ2)
On a class of fully nonlinear elliptic equations in dimension two
No full textWe study existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci's extremal operators in dimension two. In particular we prove the existence of a positive solution of a fully nonlinear version of the Liouville equation in the plane. Moreover for the Mλ,Λ− operator, we show the existence of a critical exponent and give bounds for it
Overdetermined problems and constant mean curvature surfaces in cones
No full textWe consider a partially overdetermined problem in a sector-like domain Ω in a cone Σ in RN, N ≥ 2, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that Ω is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces Γ with boundary which satisfy a ‘gluing’ condition with respect to the cone Σ. We prove that if either the cone is convex or the surface is a radial graph then Γ must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed
SYMMETRY RESULTS FOR COOPERATIVE ELLIPTIC SYSTEMS VIA LINEARIZATION
No full textIn this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in an annulus in R-N, N >= 2. More precisely we prove that solutions having Morse index j <= N are foliated Schwarz symmetric if the nonlinearity is convex and a full coupling condition is satisfied along the solution
Sectional symmetry of solutions of elliptic systems in cylindrical domains
No full textIn this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains.
The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in
cite{DaPaSys, DaGlPa1, Pa, PaWe}
- …
