This paper focuses on the following homogeneous elliptic system with critical growth -Δ_p u + V(x)|u|^p-2u = 1/p^* Q_u(u,v) inℝ^N, -Δ_p v + W(x)|v|^p-2v = 1/p^* Q_v(u,v) inℝ^N, where 1 < p < N, p* = pN/(N − p), and V, W: ℝN → ℝ are two sign-changing functions. By applying a variant of the second concentration-compactness principle, we demonstrate the existence of a mountain-pass solution for the above system. Moreover, we combine a recent global compactness result by Cintra and Correia [7] with Krasnoselskii’s genus theory to demonstrate that the system has at least N distinct pairs of non-trivial solutions in the case of small perturbations of the potentials.
In this paper, we study the characterization of inner uniformity of bounded domains G in ℝn, and prove that the following three conditions are equivalent: (1) G is inner uniform; (2) G is Gromov hyperbolic and its inner metric boundary is naturally quasisymmetrically equivalent to the Gromov boundary; (3) G is Gromov hyperbolic and linearly locally connected with respect to the inner metric. The equivalence between the conditions (1) and (2), and the implication from (2) to (3) affirmatively answer three questions raised by Bonk, Heinonen, and Koskela in 2001.
The paper investigates the well-posedness and the complete regularity of the weak solutions, and the longtime dynamics for the structurally damped wave equation with almost-linear h(x, ut) and supercritical nonlinearity g(x, u) on ℝN (N ⩾ 3): utt − Δu + (− Δ)αut + h(x, ut) + g(x, u) = f, where the perturbed parameter α ∈ (1/2, 1) is a dissipative index determining the dissipative strength. We show that when the growth order p of the nonlinearity g(x, u) is up to the supercritical range: p^∗
We are concerned with a critical Choquard system with prescribed mass -Δ u+λ_1u=(I_μ∗| u|^2_μ^∗)| u|^2_μ^∗-2u+ν p(I_μ∗| v|^q)| u|^p-2u in ℝ^N, -Δ v+λ_2v=(I_μ∗| v|^2_μ^∗)| v|^2_μ^∗-2v+ν q(I_μ∗| u|^q)| v|^p-2v in ℝ^N, ∫_ℝ^Nu^2=a^2, ∫_ℝ^Nv^2=b^2, where N ≥ 3, 0 < μ < N, ν ∈ ℝ, Iμ: ℝN → ℝ is a Riesz potential, 2_μ^∗:=2N-μN-2 and 2N-μN-2
We investigate the existence, asymptotic boundary behavior and uniqueness of viscosity solutions u ∈ C0(Ω) of equations M_𝐚(D^2u)=f(u)+h(x) in Ω ⊂ ℝn such that u(x) → ∞ as x → ∂Ω. Such solutions are referred to as large or boundary blow-up solutions. Here, Ω is a smooth bounded domain, M_𝐚 is a weighted partial trace operator, f is a non-decreasing function that satisfies the Keller–Osserman condition, and h is a continuous function in Ω. The main difficulty in the investigation rests on the possibility that M_𝐚 is very degenerate elliptic, and h is unbounded as well as sign-changing in Ω. To the best of our knowledge, large solutions to equations involving partial trace operators have not been investigated before.
