MULTIPLE POSITIVE SOLUTIONS FOR A CRITICAL GROWTH PROBLEM WITH HARDY POTENTIAL
2006 ◽
Vol 49
(1)
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pp. 53-69
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Keyword(s):
AbstractIn this paper we study the existence and nonexistence of multiple positive solutions for the Dirichlet problem:$$ -\Delta{u}-\mu\frac{u}{|x|^2}=\lambda(1+u)^p,\quad u\gt0,\quad u\in H^1_0(\varOmega), \tag{*} $$where $0\leq\mu\lt(\frac{1}{2}(N-2))^2$, $\lambda\gt0$, $1\ltp\leq(N+2)/(N-2)$, $N\geq3$. Using the sub–supersolution method and the variational approach, we prove that there exists a positive number $\lambda^*$ such that problem (*) possesses at least two positive solutions if $\lambda\in(0,\lambda^*)$, a unique positive solution if $\lambda=\lambda^*$, and no positive solution if $\lambda\in(\lambda^*,\infty)$.
2015 ◽
Vol 66
(5)
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pp. 2441-2471
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Multiple positive solutions for a nonlinear Dirichlet problem with non-convex vector-valued response
2005 ◽
Vol 135
(1)
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pp. 105-117
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2003 ◽
Vol 67
(3)
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pp. 413-427
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2015 ◽
Vol 145
(2)
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pp. 365-390
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