| Abstract: |
| In this talk we explore the effect of the presence of a nonnegative heterogeneous weight function $a(x)$ on the existence of solutions to semilinear elliptic equations with critical nonlinearity a la Trudinger-Moser:
$$-\Delta u=a(x)h(u)e^{\alpha u^2}\, \mbox{in}\, \Omega\quad u=0 \, \mbox{on}\, \partial\Omega $$
in dimension 2, i.e. when $\Omega $ is a smooth bounded domain in $\mathbb{R}^2$. We consider this under the assumption that either $a(x)$ grows sufficiently fast at an interior point of $\Omega$ or is radially symmetric and rapidly vanishing weight function. We are then able to prove existence of positive or sign changing solutions {\it without the usual growth restriction on the lower order term of critical nonlinearities (that is, $h(u)$)}. By highlighting the effect of the heterogeneity in relaxing growth restrictions on the lower order terms, our results complement and generalize a number of existing results in the literature of such problems. |
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