N- Laplacian problems with critical double exponential nonlinearities

Abstract

In this paper, we prove the existence of a nontrivial solution for the following boundary value problem    −div ω(x)|∇u(x)| N −2 ∇u(x) = f (x, u), in B;

Non-autonomous weighted elliptic equations with double exponential growth

“…where B 1 is the unit disk of R 2 , w(x) = log e |x| and the nonlinearities f are of double exponential growth, was studied in [15]. Recently, Deng, Hu and Tang [18] studied the following problem…”

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

“…Calanchi, B. Ruf and F. Sani proved in [18] the existence of a nontrivial radial solution for the case N = 2. This result has been recently generalized by C. Zhang in [43] and by S. Deng, T. Hu and C-L. Tang in [22] for higher dimensions.…”

“…Here, once again, the presence of the singularity plays an essential role. This essential role can also be clearly seen in the third point of difference between the singular case (treated in the present work) and the nonsingular one (treated in [12]), which is the sharpness of the inequality (22). In fact, the inequality (1.20) in [12] is not necessarily sharp despite our effort to give the best range where the supremum is finite.…”

“…This fact illustrates in a very clear way the capital role played by the singularity to recover this sharpness property. Finally, as it was done in [12], we provide an improvement of ( 18) and ( 22) for weakly convergent sequences in E β , 0 < β ≤ 1 with a constant larger than those found in (18) and (22). These results are new and are very useful to study some singular nonlinear equations involving new exponential growth conditions at infinity.…”

“…By (44), we have Combining ( 46) and ( 43), we easily see that ( 21) holds. Now, we move to establish (22). For that aim, we first note that by (43), we have…”