Sobolev embeddings involving symmetry

Existence of positive solutions for a class of semilinear and quasilinear elliptic equations with supercritical case

“…where 0 < q < 1 < p < 2 * − 1 + τ , and λ, τ are positive real parameters (τ is the constant obtained in [51]). Let h(x) satisfy the following conditions:…”

“…Recently, Wenzhi Wang [51] establishes embedding results (see Lemma 2.1) in a cylindrically symmetric domain. He proves that functions having such symmetry and belonging to H 1 0 can be embedded compactly into some weighted L p spaces, with p superior to the critical Sobolev exponent.…”

“…Our purpose is to use the embedding theorem in [51] and classical variational tools to solve the problem with supercritical growth. This paper is organized as follows.…”

Solutions of elliptic problems of p-Laplacian type in a cylindrical symmetric domain

“…Motivated by the recent results on the effect of the topology of the domain in [5,6,20], in the present work, we will investigate (1.1) with condition (1.3) in the case when Ω = Ω 1 × Ω 2 ⊂ R N is a bounded domain having cylindrical symmetry, Ω 1 ⊂ R m is a bounded regular domain and Ω 2 is a k-dimensional ball of radius R, centered in the origin and m + k = N , and m 1, k 2. Using variational methods, we prove some existence and multiplicity results for (1.1) in the critical and supercritical cases q p .…”

Compact Sobolev embedding theorems involving symmetry and its application

“…Generally speaking, some kinds of the geometric and topological properties of the domain lead to the solvability of elliptic problems. We note that symmetry can improve Sobolev embeddings (see [1][2][3][4] and references therein). If Ω = R N , P. L. Lions (see [1,2]) showed that the embedding W In recent years, there has been an increasing interest in the study of variational problems with variable exponent (see [5][6][7][8][9]).…”