| Abstract: |
| In this talk we will discuss a compactness result for the following class of subcritical/critical problems
\[
-\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{p_n-2}u \text{ in } \Omega,\quad u=0 \text{ in } \mathbb{R}^N \setminus \Omega,
\]
where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $N\geq3$, $s\in(0,1)$, $\lambda>0$, $p_n \in (p,2^*]$ and $p_n\to 2^*=\frac{2N}{N-2}$. Specifically for
\[
p\in
\begin{array}{l}
\left(2+\dfrac{4s}{N-2},2^*\right), \text{ if } N>6-4s; \
\left(2^*-1,2^*\right), \text{ if } N\leq 6-4s,
\end{array}
\]
we prove that if $\{u_n\}$ is any sequence of solution to the above equation with $\|u_n\|_{X_0}\leq C$ in some appropriate Sobolev space $X_0$ then $u_n$ strongly converges in $X_0$. To the best of our knowledge, this is the first paper to address this type of compactness result for a non-homogeneous operator. Due to the presence of the non-homogeneous operator, our proof requires a non-trivial adaptation of the methods developed by Devillanova and Solimini (Adv. Differential Equations, 2002) and Yan, Yang, and Yu (J. Funct. Anal., 2015). An application of this compactness result, under the same ranges of $N$ and $p$, we establish infinitely many sign-changing solutions to the following Brezis-Nirenberg type problem:
\[
-\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{2^*-2}u \text{ in } \Omega,\quad u=0 \text{ in } \mathbb{R}^N \setminus \Omega.
\] |
|