Nonexistence of nodal solutions of elliptic equations with critical growth in $\mathbb {R}^2$

Abstract:

Let $f(t) = h(t){e^{b{t^2}}}$ be a function of critical growth. Under a suitable assumption on $h$, we prove that \[ \begin {array}{*{20}{c}} { - \Delta u = f(u)} \hfill & {{\text {in}}\;B(R) \subset {\mathbb {R}^2},} \hfill \\ {u = 0} \hfill & {{\text {on}}\;\partial B(R),} \hfill \\ \end {array} \] does not admit a radial solution which changes sign for sufficiently small $R$.