A new Singular Trudinger–Moser Type Inequality with Logarithmic Weights and Applications

Abstract

In this paper, we establish a new singular Trudinger–Moser type inequality for radial Sobolev spaces with logarithmic weights. The existence of nontrivial solutions is proved for an elliptic equation defined in {\mathbb{R}^{n}}, relying on variational methods and involving a nonlinearity with doubly exponential growth at infinity.

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

“…Clearly, when σ = 0, we obtain an extension of Theorem 1.5 to the whole space R N . This result with those established in [9,10,11] are the first attempts to extend the inequalities established by M. Calanchi and B. Ruf in Theorem 1.3, Theorem 1.4, Theorem 1.5 and Theorem 1.6 to the whole space R N . For the value α α N,β + σ N = 1, we could not prove or disprove that the supremum in (11) is finite.…”

“…Furthermore, in [10] we extended the last singular inequalities to the whole euclidean space R N , N ≥ 2. In fact, we considered a radial weight w β defined by…”

“…For that space, we obtained the following sharp singular Trudinger-Moser inequality which can be considered as a second extension of Theorem 1.2: Theorem 1.9. Let 0 < β < 1, 0 < σ < N and w β be defined by (10). For all u ∈ Y ′ β , we have…”

“…The value β = 1 is a kind of second order limiting case. In [10], we established the following extension of Theorem 1.6: Theorem 1.10.…”

“…The first result in the present work is the following extension of Theorem 1.2: Theorem 1.11. Let 0 < σ < N, 0 < β < 1 and w β be defined by (10).…”