A Trudinger--Moser inequality in a weighted Sobolev space and applications (Q2922200)

The authors establish a Trudinger-Moser inequality in a weighted Sobolev space, precisely NEWLINE\[NEWLINE \int_{\mathbb R^2}K(x)|u|^2(e^{\beta u^2}-1)dx\leq C(M,\beta), NEWLINE\]NEWLINE for any \(u\) in the closure of \(C_c^\infty(\mathbb R^2)\) with respect to the weighted norm in \(L^p_K(\mathbb R^2)\). The inequality is applied in the study of the elliptic equation NEWLINE\[NEWLINE -\text{div}\, (K(x)\nabla u)=K(x) f(u)+h \qquad \text{ in } \quad \mathbb R^2\,, NEWLINE\]NEWLINE where \(K(x)=\exp(|x|^2/4),\) \(f\) has exponential critical growth and \(h\) belongs to the dual of an appropriate function space. The authors prove that the problem has at least two solutions provided \(h\not=0\) is small.