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Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation

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  • Some critical Sobolev-Hardy inequalities with weight of distance function $d^{\frac{\alpha}{p}p^*}$ are established in a bounded domain $\Omega$, where $d$ is the distance to the boundary $\partial\Omega$. Using these inequalities we get the result that the embedding $\mathcal{D}^{1, 2}(\Omega, d^\alpha)\hookrightarrow L^q(\Omega, d^{\beta})$ is compact if $1\leq q<2^*$ and $\beta >\frac{\alpha}{2}q+\frac{q}{2^*}-1$. By the compactness result and critical-point theory about sign-changing solutions, we obtain infinitely many sign-changing solutions to a degenerate Dirichlet elliptic equation $-\hbox{div}(d^\alpha \nabla u)- \frac{(1-\alpha )^2}{4} d^{\alpha-2} u=f(x,u)$ provided that $f(x,u)$ satisfies suitable conditions.
    Mathematics Subject Classification: Primary: 35J57, 35J66; Secondary: 35J75.


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