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Four solutions of an elliptic problem with double singularity

Eugénio Rocha, Jianqing Chen, Kelly Murillo


We consider the existence of nontrivialsolutions $u\in H_0^1(\Omega)$ of a Dirichlet problem with equation$$-\Delta u -\frac{\mu}{|x|^2}u = \lambda f(x)|u|^{q-2}u + \frac{|u|^{\frac{4-2s}{N-2}}u}{|x|^s},\quad x\in\Omega\backslash\{0\},$$where $0\in\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary, $f:\Omega\rightarrow \mathbb{R}$ is a real continuous and positivefunction, $0\leq s<2$,$0\leq \mu < \bar{\mu} -4$ (necessarily $N> 6$), and$\frac{N+\sqrt{{\bar{\mu}-\mu}}}{\sqrt{\bar{\mu}} +\sqrt{\bar{\mu}-\mu}} < q < 2$, with $\bar{\mu} = (N-2)^2/4$. Note that $\frac{4-2s}{N-2}$ is the critical Sobolev-Hardy exponent minus 2. By variational methods we show that there is (at least) twopositive solutions and (at least) one pair of sign-changing solutionsin $H_0^1(\Omega)$ for any $\lambda\in (0,\Lambda^*)$, where$\Lambda^*$ is a positive value suitable defined.


Elliptic equations; Positive solutions; Sign changing solutions; Variational methods; Concave term; Singular term

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