2017/03/07 by Clapp, Monica, Rizzi, Matteo
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1703.02257
We study the problem% -Δv+λv=| v| p-2v in Ω,\qquadv=0 on ∂Ω, % for λ∈ℝ and supercritical exponents p, in domains of the form% Ω:=\(y,z)∈ℝN-m-1×ℝm+1:(y,| z| )∈Θ\, where m≥1, N-m≥3, and Θ is a bounded domain in ℝ% N-m whose closure is contained in ℝN-m-1×(0,∞). Under some symmetry assumptions on Θ, we show that this problem has infinitely many solutions for every λ in an interval which contains [0,∞) and p>2 up to some number which is larger than the (m+1)st critical exponent 2N,m∗:=(2(N-m))/(N-m-2). We also exhibit domains with a shrinking hole, in which there are a positive and a nodal solution which concentrate on a sphere, developing a single layer that blows up at an m-dimensional sphere contained in the boundary of Ω, as the hole shrinks and p→2N,m∗ from above. The limit profile of the positive solution, in the transversal direction to the sphere of concentration, is a rescaling of the standard bubble, whereas that of the nodal solution is a rescaling of a nonradial sign-changing solution to the problem% -Δu=| u| ^2n∗-2u,\qquadu∈ D1,2(ℝn), where 2n∗:=(2n)/(n-2) is the critical exponent in dimension n.\medskip