We study a critical bi-nonlocal anisotropic Schrödinger–Kirchhoff problem with variable exponents in a bounded Lipschitz domain of dimension $ N\ge2 $. The equation combines a Kirchhoff coefficient depending on the full anisotropic modular, a second nonlocal reaction of the form $ \bigl(\int_\Omega F(x, u)\, dx\bigr)^r f(x, u) $, and a reaction exponent that may reach the $ p_m(\cdot) $-based critical threshold in the interior of the domain while remaining subcritical on the boundary. We first establish the variational framework for general anisotropic Leray–Lions potentials with directional growth $ 1 < p_i^-\le p_i(x)\le p_i^+ < N $. The anisotropic concentration-compactness principle records the possible critical defect measures. Independently, testing the derivative with the weak difference yields a global monotonicity defect identity; the subcritical nonlocal source vanishes, the critical term contributes a non-negative defect, and the $ (S_+) $ property of the principal operator gives strong convergence. This convergence then identifies the recorded measures and eliminates every concentration mass. We then introduce a truncated even functional and apply Krasnosel'skii's genus theory to obtain infinitely many distinct pairs of nontrivial weak solutions for sufficiently small values of the parameter. The abstract assumptions are verified for standard anisotropic $ p_i(x) $-growth operators, explicit power-type sources, and both polynomial and bounded Kirchhoff laws.
Citation: Oussama Sebbar, Nabil Chems Eddine, Zine El Abidine Guennoun. Compactness and multiplicity for a critical bi-nonlocal anisotropic Schrödinger–Kirchhoff problem with variable exponents[J]. Communications in Analysis and Mechanics, 2026, 18(4): 889-930. doi: 10.3934/cam.2026035
We study a critical bi-nonlocal anisotropic Schrödinger–Kirchhoff problem with variable exponents in a bounded Lipschitz domain of dimension $ N\ge2 $. The equation combines a Kirchhoff coefficient depending on the full anisotropic modular, a second nonlocal reaction of the form $ \bigl(\int_\Omega F(x, u)\, dx\bigr)^r f(x, u) $, and a reaction exponent that may reach the $ p_m(\cdot) $-based critical threshold in the interior of the domain while remaining subcritical on the boundary. We first establish the variational framework for general anisotropic Leray–Lions potentials with directional growth $ 1 < p_i^-\le p_i(x)\le p_i^+ < N $. The anisotropic concentration-compactness principle records the possible critical defect measures. Independently, testing the derivative with the weak difference yields a global monotonicity defect identity; the subcritical nonlocal source vanishes, the critical term contributes a non-negative defect, and the $ (S_+) $ property of the principal operator gives strong convergence. This convergence then identifies the recorded measures and eliminates every concentration mass. We then introduce a truncated even functional and apply Krasnosel'skii's genus theory to obtain infinitely many distinct pairs of nontrivial weak solutions for sufficiently small values of the parameter. The abstract assumptions are verified for standard anisotropic $ p_i(x) $-growth operators, explicit power-type sources, and both polynomial and bounded Kirchhoff laws.
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Oussama Sebbar, Nabil Chems Eddine, Zine El Abidine Guennoun. Compactness and multiplicity for a critical bi-nonlocal anisotropic Schrödinger–Kirchhoff problem with variable exponents[J]. Communications in Analysis and Mechanics, 2026, 18(4): 889-930. doi: 10.3934/cam.2026035
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