2 results
On a perturbed critical semilinear equation with singularity
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 27 February 2025, pp. 1-35
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In this article, we investigate necessary and sufficient conditions on the perturbation ρ for the existence of positive least energy solutions of the critical singular semilinear elliptic equation
$ -\Delta u = \frac{|u|^{2^{*}(s)-2}}{|x|^s}u + \rho(u) $ with Dirichlet boundary condition in a bounded smooth domain in 
$\mathbb R^n$ containing the origin, where 
$2^*(s)=\frac{2(n-s)}{n-2}$, 
$0\leq s \lt 2 \lt n$. We show that the almost necessary and sufficient condition obtained for the case s = 0 in [1] differs conceptually when 
$0 \lt s \lt 2$.
The existence of a ground-state solution for a class of Kirchhoff-type equations in ℝN
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 146 / Issue 2 / April 2016
- Published online by Cambridge University Press:
- 03 March 2016, pp. 371-391
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- April 2016
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In this paper, we study the following Kirchhoff-type equation:
where a, b are positive constants and N = 1, 2, 3. Under appropriate assumptions on V, K and g, we obtain a ground-state solution by using the approach developed by Szulkin and Weth in 2010.
