Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains

In this paper we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates (r, θ, z) , we investigate the shape of possibly sign-changing solutions whose derivative in θ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of θ or strictly monotone with respect to θ. In the special case of a planar domain, the result holds in a circular sector as well as in an annular one, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case where its opening is larger than π

Quasi-radial solutions for the Lane–Emden problem in the ball

We consider the Lane-Emden problem in the unit ball B of ℝ^2 centered at the origin with Dirichlet boundary conditions and exponent ∈(1,+∞) of the power nonlinearity. We prove the existence of sign-changing solutions having 2 nodal domains, whose nodal line does not touch ∂ and which are non-radial. We call these solutions quasi-radial. The result is obtained for any p sufficiently large, considering least energy nodal solutions in spaces of functions invariant under suitable dihedral groups of symmetry and proving that they fulfill the required qualitative properties. We also show that these symmetric least energy solutions are instead radial for p close enough to 1, thus displaying a breaking of symmetry phenomenon in dependence on the exponent p. We then investigate the nonradial bifurcation at certain values of p from the sign-changing radial least energy solution of.. The bifurcation result gives again, with a different approach and for values of p close to the ones at which the bifurcations appear, the existence of non-radial but quasi-radial nodal solutions

Qualitative analysis on the critical points of the Robin function

Let \O\subset\R^N be a smooth bounded domain with $N\ge2$ and \O_\e=\O\backslash B(P,\e) where B(P,\e) is the ball centered at P\in\O and radius \e. In this paper, we establish the number, location and non-degeneracy of critical points of the Robin function in \O_\e for \e small enough. We will show that the location of $P$ plays a crucial role on the existence and multiplicity of the critical points. The proof of our result is a consequence of delicate estimates on the Green function near to \partial B(P,\e). Some applications to compute the exact number of solutions of related well-studied nonlinear elliptic problems will be showed

Bifurcation and asymptotic analysis for a class of supercritical elliptic problems in an exterior domain

We 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

EXISTENCE AND MULTIPLICITY RESULTS FOR EQUATIONS WITH NEARLY CRITICAL GROWTH

We consider the problem -Delta u = K(x)u(p epsilon) in R(n) u > 0 in R(n) where p = n+2/n-2, p(epsilon) = p - epsilon, n >= 3; epsilon > 0 and K (x) > 0 in R(n). We prove an existence and multiplicity result for single peaked solutions of our problem concentrating at a fixed critical point of K (x) and some other related results

Symmetry results for cooperative elliptic systems in unbounded domains

In this paper we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in $\R^N$, $N\geq 2$ 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

On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's (preprint)

We investigate nodal radial solutions to semilinear problems of type {−Δu=f(|x|,u) in Ω,u=0 on ∂Ω, where Ω is a bounded radially symmetric domain of RN (N≥2) and f is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, and describe the symmetries of the eigenfunctions. Next we use this characterization to give a lower bound for the Morse index; in such a way we give an alternative proof of an already known estimate for the autonomous problem and we furnish a new estimate for H\'enon type problems with f(|x|,u)=|x|αf(u). Concerning the real H\'enon problem, f(|x|,u)=|x|α|u|p−1u, we prove radial nondegeneracy and show that the radial Morse index is equal to the number of nodal zones

A non-variational system involving the critical Sobolev exponent. The radial case

In this paper we consider the non-variational system−Δui=Σj=1kaijujN+2N−2inRN,ui>0inRN,ui∈D1,2(RN),and we give some sufficient conditions on thematrix (aij)i, j =1,..,k which ensure the existence of solutions bifurcating from the bubble of the critical Sobolev equation