| Abstract: |
| This is a continuation of our multiplicity results for elliptic PDEs with heterogeneous critical nonlinearities \`a la Trudinger-Moser. Our results are ultimately connected to the increase in the threshold of compactness that can be achieved in the Trudinger-Moser inequality under radial symmetry and in the presence of a rapidly vanishing radial weight function in $H^1_{0,r}(B_R)$. They are generalizations to dimension $N=2$ of the 1981 seminal result of W.-M.-Ni in dimension $N\geq 3$.
In particular, we can prove the existence of a positive, a negative, and a sign-changing solution for the equation
\begin{equation}
-\Delta u = a(r)h(u) e^{alpha u^2}\ in \,\,\, B(R_1,R_2)\,,\ \ on\ \partial B(R_1,R_2)\,
\end{equation}
without requiring the oddness assumption on the nonlinearity and, in fact, we can prove the existence of {\it Infinitely Many Sign-Changing Solutions) in the case of a ball. All of our results are obtained without any growth restriction on the lower-order terms of the nonlinearity and, to our knowledge, generalize most existing results in the literature of such problems. |
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