Existence of Solutions for a Mixed Local and Nonlocal Elliptic Problem in \(\mathbb{R}^N\)
Yunchuan Dai *
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. China.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we investigate the existence of solutions to the following equation:
−Δu + (−Δ)su = λ|u|p−2u + μ|u|q−2u in \(\mathbb{R}^N\),
where N ≥ 3, λ ≥ 0, μ ≥ 0, μ + λ > 0, 0 < s < 1, and 2∗s ≤ p < q ≤ 2∗. Here, \(2^*_s\) = \(\frac{2N}{N-2s}\) and 2∗ = \(\frac{2N}{N-2}\) denote the fractional and Sobolev critical exponents, respectively. This study fills a theoretical gap in the variational framework for mixed operators by overcoming the loss of compactness caused by critical terms. We analyze three distinct scenarios regarding the parameters p, q, λ, and μ. By combining the mountain pass theorem with Lions’ lemma and the principle of concentration compactness, we establish the existence of a nontrivial solution for each case.
Keywords: Mixed local and nonlocal operators, sobolev inequality, principle of concentration compactness