We investigated the existence of boundary blow-up solutions for a slightly subcritical semilinear elliptic problem posed in a smooth bounded domain of $ \mathbb{R}^n $ with $ n\in\{4, 5, 6\} $. We constructed solutions that have a nonzero weak limit while simultaneously blowing up at a boundary point. This behavior stands in contrast to the well-known compactness properties on manifolds without boundary for small dimensions, where such weak convergence would force strong convergence. Our construction shows that, in low dimensions, the presence of a boundary allows blow-up and residual mass to coexist. Moreover, we identified the precise boundary points where this concentration occurs. The results are proved by means of delicate asymptotic estimates of the gradient of the associated Euler–Lagrange functional.
Citation: Dalal Almutairi, Mohamed Ben Ayed. Boundary peak solutions with residual mass of a slightly subcritical Hénon type problem[J]. AIMS Mathematics, 2026, 11(6): 15324-15360. doi: 10.3934/math.2026630
We investigated the existence of boundary blow-up solutions for a slightly subcritical semilinear elliptic problem posed in a smooth bounded domain of $ \mathbb{R}^n $ with $ n\in\{4, 5, 6\} $. We constructed solutions that have a nonzero weak limit while simultaneously blowing up at a boundary point. This behavior stands in contrast to the well-known compactness properties on manifolds without boundary for small dimensions, where such weak convergence would force strong convergence. Our construction shows that, in low dimensions, the presence of a boundary allows blow-up and residual mass to coexist. Moreover, we identified the precise boundary points where this concentration occurs. The results are proved by means of delicate asymptotic estimates of the gradient of the associated Euler–Lagrange functional.
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Dalal Almutairi, Mohamed Ben Ayed. Boundary peak solutions with residual mass of a slightly subcritical Hénon type problem[J]. AIMS Mathematics, 2026, 11(6): 15324-15360. doi: 10.3934/math.2026630
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