Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth

By Nassif Ghoussoub, F. Robert. 2005

81 citations recorded by the source corpus.

We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $\Omega$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial \Omega$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $\Omega$ is star-shaped around 0). However, in contrast to the non-satisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of $\partial \Omega$ at 0 in at least one direction. More precisely, we need the principal curvatures of $\partial \Omega$ at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of $\partial \Omega$ at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0f1433810c44b8f681e07da875ac328d49d27873

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form (2014, background): “For instance, the attainability for the best constant of the Caffarelli–Kohn–Nirenberg inequality (1.6) was extensively studied, see e.g. Chern and Lin [7], Ghoussoub and Kang [9], Ghoussoub and Robert [10,11] and Hsia, Lin and Wadade [12], where the authors proved the existence of a maximizer assoc”
Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary (2010, background): “If a = 0 and 1 > b > 0, then the result was proved by Ghoussoub and Robert [12].”
Hardy—Poincaré inequalities with boundary singularities (2010, background): “1, by i) and by the existence theorems proved in [8], [9] and [10] for a related superlinear problem.”
Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights (2008, background): “[2,10,12,18,23,42], and for general p > 1 see [17,20,24,50]; while in unbounded domains and for p = 2 cf.”