Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Abstract

ABSTRACT. Assuming B R is a ball in R N , we analyze the positive solutions of the problemthat branch out from the constant solution u = 1 as p grows from 2 to +∞. The non-zero constant positive solution is the unique positive solution for p close to 2. We show that there exist arbitrarily many positive solutions as p → ∞ (in particular, for supercritical exponents) or as R → ∞ for any fixed value of p > 2, answering partially a conjecture in [12]. We give the explicit lower bounds for p and R so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.

Singular radial solutions for the Keller-Segel equation in high dimension

“…In this form it is more visible that the nonlinearity becomes 'more critical' if µ is large, which is equivalent to λ being small. The following bifurcation result for (1.6) with parameter µ was obtained in [3], see [6] for an analogous result for (1.4). Note that for fixed parameters the radial solutions of the second order equations are uniquely determined by the value of the function at 0 (since u ′ (0) = 0), therefore it suffices to investigate bifurcation diagrams in R 2 with coordinates corresponding to µ and u(0).…”

A nonlocal supercritical Neumann problem

“…Some further comments on the condition (f 3 ) and its variants are now in order. In the local setting, it was first conjectured in [4] and then proved in [3,12,5] that if f ′ (u 0 ) satisfies f ′ (u 0 ) > 1 + λ rad k+1 (R) for some k ≥ 1, (5.9) where λ rad k+1 (R) is the (k + 1)-st radial eigenvalue of the Neumann Laplacian in B R , then the Neumann problem −∆u + u = f (u) in B R admits a radial positive solution having exactly k intersections with the constant u 0 . It would be interesting to prove a similar result also in this fractional setting.…”

A Paneitz–Branson type equation with Neumann boundary conditions

“…Using Poincaré inequality and the uniform convergence to zero, one then shows that u − 1 |Ω| Ω u dx = 0 when α is small enough. This is the original argument of Ni and Takagi [48], see also [5,13].…”

“…The bound is clearly sharp. In the subcritical case, one can use elliptic regularity to bootstrap the corresponding estimate to get a better bound or use Gidas-Spruck blow-up technique [28] to show directly that u converges uniformly to zero as α → 0, see for instance [13,48]. In the critical case, we need a further hypothesis to improve the bound as shown by the next lemma.…”