2026/05/24 by Mousomi Bhakta, Nirjan Biswas, Paramananda Das
#math.AP
In this paper, we consider the following mixed local nonlocal Brezis-Nirenberg problem P2^* -Δu+(-Δ)s u=λ|u|p-2u+|u|2^*-2u in Ω, u=0 in ℝN ∖ Ω, where Ω⊂ℝN is a smooth bounded domain, N≥3, s∈(0,1), λ>0, and 2≤ p<2^*=(2N)/(N-2). We establish a compactness result for the following class of subcritical/critical problems Ppn -Δu+(-Δ)s u=λ|u|p-2u+|u|pn-2u in Ω, u=0 in ℝN ∖ Ω, where pn ∈ (p,2^* ] and pn→ 2^*. Specifically, for p ∈ (2+(4s)/(N-2),2^*) when N>6-4s, and for p ∈ (2^*-1,2^*) when N≤6-4s, we prove that any bounded sequence of solutions \un\ to \eqrefsubproabstract is relatively compact in the energy space, and converges strongly to a nontrivial solution to \eqrefcritproabstract. 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, proving the compactness result requires several delicate new and novel estimates, which we believe will be of independent interest for further studies of related problems. As an application of this compactness result, under the same ranges of N and p, we prove that \eqrefcritproabstract admits infinitely many sign-changing solutions.