Hardy–Sobolev–Maz'ya type equations in bounded domains

Abstract

We study the regularity, Palais-Smale characterization and existence/nonexistence of solutions of the Hardy-Sobolev-Maz'yashow different behaviors of PS sequences depending on t = 0 or t > 0.

Poincaré–Sobolev equations in the hyperbolic space

“…Classification of PS sequences has been done for various problems in bounded domains in R N and on compact Riemannian manifolds, where the lack of compactness is due to the concentration phenomenon (See [4,13,15],... and the references therein). However the present case should be compared with the case of infinite volume case, say the critical equations in R N .…”

“…A point x ∈ R n is denoted as x = (y, z) ∈ R k × R n−k . One can see that u ∈ D 1,2 (R n ) is a cylindrically symmetric solutions of (1.2) (i.e., u(x) =ũ(|y|, z)) iff v = w • M solves (1.1) with dimension N = n − k + 1, p = p t and λ = η + (n−k) 2 −(k−2) 2 4 where w(r, z) = r n−2 2ũ (r, z) for (r, z) ∈ (0, ∞) × R n−k and M : B n−k+1 → (0, ∞) × R n−k is the standard isometry (see (6.29)) between the B N and the upper half space model of the hyperbolic space (see [8,9] for details). Note that when k = 2 and η = 0, then we have λ = ( N −1 2 ) 2 in (1.1).…”

Existence of Multiple Solutions for P-Laplacian Problems Involving Critical Exponents and Singular Cylindrical Potential

“…In case p>2 and 1< k < N , equations with cylindrical potentials were also studied by many people [1,4,[7][8][9][10]. For instance, in [11], Xuan studied the multiple weak solutions for p-Laplace equation with singularity and cylindrical symmetry in bounded domains.…”

Five Nontrivial Solutions of p-Laplacian Problems Involving Critical Exponents and Singular Cylindrical Potential

“…In the case of 1 < k < N, equations with cylindrical potentials were also studied by many people [8][9][10][11][12][13][14]. For instance, in [15], Xuan studied the multiple weak solutions for p-Laplace equation with singularity and cylindrical symmetry in bounded domains.…”