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A weighted Trudinger–Moser inequalities and applications to some weighted $(N,q)-$
Laplacian equation in $\mathbb {R}^N$
with new exponential growth conditions
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 07 September 2023, pp. 1-52
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In this paper, we prove some weighted sharp inequalities of Trudinger–Moser type. The weights considered here have a logarithmic growth. These inequalities are completely new and are established in some new Sobolev spaces where the norm is a mixture of the norm of the gradient in two different Lebesgue spaces. This fact allowed us to prove a very interesting result of sharpness for the case of doubly exponential growth at infinity. Some improvements of these inequalities for the weakly convergent sequences are also proved using a version of the Concentration-Compactness principle of P.L. Lions. Taking profit of these inequalities, we treat in the last part of this work some elliptic quasilinear equation involving the weighted $(N,q)-$
Laplacian operator where $1 < q < N$
and a nonlinearities enjoying a new type of exponential growth condition at infinity.
