Ground states of critical and supercritical problems of Brezis–Nirenberg type

Abstract

Abstract. We study the existence of symmetric ground states to the supercritical problemin a domain of the formwhere Θ is a bounded smooth domain such thatis the (k + 1)-st critical exponent. We show that symmetric ground states exist for λ in some interval to the left of each symmetric eigenvalue, and that no symmetric ground states exist in some interval (−∞, λ * ) with λ * > 0 if k ≥ 2.Related to this question is the existence of ground states to the anisotropic critical problemwhere a, b, c are positive continuous functions on Θ. We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of λ, and obtain a bifurcation result for ground states.

The Brezis–Nirenberg problem for the curl–curl operator

“…Namely, we look for solutions of the following equation (5.1) ∇ × µ(x) −1 ∇ × E − V (x)E = f (x, E) in Ω together with boundary condition (1.4). Recall that the anisotropic Brezis-Nirenberg-type variant of (1.6) has been recently studied by Clapp, Pistoia and Szulkin [16]. In (5.1), the permeability tensor µ(x) is of the form (5.2)…”

Numerical solution of a critical Sobolev exponent problem with weight on 𝕊3

“…The partial differential equations are one of the most celebrate tools discovered from modelling many phenomena in nature. The differential equation in (1) can be a laboratory of finding many methods to deal with similar mathematical models which arise in different branch of sciences [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The problem (1) is interesting to study and has different new features, since this model is the stationary equation of convection-diffusion models appearing frequently in connection with conservation laws.…”