Existence of solution for a singular elliptic equation with critical Sobolev-Hardy exponents.

Li, Juan

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We consider local minimizers $u : ℝ^{2} \supset Ω \to ℝ^{N}$ of variational integrals like $\int_{Ω} [(1 + | \partial_{1} {u |}^{2})^{p / 2} + (1 + | \partial_{2} {u |}^{2})^{q / 2}] d x$ or its degenerate variant $\int_{Ω} [| \partial_{1} {u |}^{p} + | \partial_{2} u |^{q}] d x$ with exponents $2 \leq p < q < \infty$ which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. (2003), 177–186. We prove interior $C^{1, α}$ - respectively $C^{1}$ -regularity of $u$ under the condition that $q < 2 p$ . For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. (2006), 349–362. ...

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