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Trudinger-Moser inequalities involving fast growth and weights with strong vanishing at zero. (English) Zbl 1341.35080

Summary: In this paper we study some weighted Trudinger-Moser type problems, namely \[ s_{F,h} = \sup\limits_{u \in H,\| u\| _H =1}\int_B F(u) h(| x|) dx, \] where \( B\subset \mathbb R^2\) represents the open unit ball centered at zero in \(\mathbb R^2\) and \(H\) stands either for \(H^1_{0, \mathrm{rad}}(B)\) or \( H^1_{\mathrm{rad}}(B)\). We present the precise balance between \(h(r)\) and \(F(t)\) that guarantees \(s_{F,h}\) to be finite. We prove that \(s_{F,h}\) is attained up to the \(h(r)\)-radially critical case. In particular, we solve two open problems in the critical growth case.

MSC:

35J86 Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators
35J15 Second-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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