Abstract
Motivated by Ruf–Saniʼs recent work, we prove an Adams type inequality and a singular Adams type inequality in the whole four-dimensional Euclidean space. As applications of those inequalities, a class of elliptic partial differential equations are considered. Existence of nontrivial weak solutions and multiplicity results are obtained via the mountain-pass theorem and the Ekelandʼs variational principle. This is a continuation of our previous work about singular Trudinger–Moser type inequality.
Highlights
Introduction and main resultsLet Ω ⊂ Rn be a smooth bounded domain
For all α ≤ αn = nω1n/−(1n−1), where ωn−1 is the area of the unit sphere in Rn. (1.1) is sharp in the sense that for any α > αn, the integrals in (1.1) are still finite, but the supremum of the integrals are infinite. (1.1) plays an essential role in the study of the following partial differential equations
U∈W2,2(R4), R4 (−∆u+u)2dx≤1 R4 (1.8). This inequality is sharp, i.e. if 32π2 is replaced by any α > 32π2, the supremum is infinite. They obtained more in [32], but here we focus on four dimensional case
Summary
Let Ω ⊂ Rn be a smooth bounded domain. The classical Trudinger-Moser inequality [26, 29, 36] says sup u∈W01,n (Ω), ∇u Ln (Ω)≤1 n eα|u| n−1 d x < ∞. To get a Trudinger-Moser type inequality in this case, D. Sani [32] were able to establish the corresponding Adams type inequality in R4, say Theorem A (Ruf-Sani). This inequality is sharp, i.e. if 32π2 is replaced by any α > 32π2, the supremum is infinite They obtained more in [32], but here we focus on four dimensional case. Theorem 1.1 can be applied to study the existence of weak solutions to the following nonlinear equation. And throughout this paper we assume 0 ≤ β < 4, a(x), b(x) are two continuous functions satisfying (A1) there exist two positive constants a0 and b0 such that a(x) ≥ a0 and b(x) ≥ b0 for all x ∈ R4;.
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