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- W4285069811 abstract "For $N geq 5$ and $a>0$, we consider the following semipositone problem begin{align*} Delta^2 u= g(x)f_a(u) text { in } mathbb{R}^N, , text{ and } , u in mathcal{D}^{2,2}(mathbb{R}^N), qquad quad mathrm{(SP)} end{align*} where $g in L^1_{loc}(mathbb{R}^N)$ is an indefinite weight function, $f_a:mathbb{R} to mathbb{R}$ is a continuous function that satisfies $f_a(t)=-a$ for $t in mathbb{R}^-$, and $mathcal{D}^{2,2}(mathbb{R}^N)$ is the completion of $mathcal{C}_c^{infty}(mathbb{R}^N)$ with respect to $(int_{mathbb{R}^N} (Delta u)^2)^{1/2}$. For $f_a$ satisfying subcritical nonlinearity and a weaker Ambrosetti-Rabinowitz type growth condition, we find the existence of $a_1>0$ such that for each $a in (0,a_1)$, (SP) admits a mountain pass solution. Further, we show that the mountain pass solution is positive if $a$ is near zero. For the positivity, we derive uniform regularity estimates of the solutions of (SP) for certain ranges in $(0,a_1)$, relying on the Riesz potential of the biharmonic operator." @default.
- W4285069811 created "2022-07-13" @default.
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- W4285069811 date "2022-07-10" @default.
- W4285069811 modified "2023-09-26" @default.
- W4285069811 title "On the fourth order semipositone problem in $mathbb{R}^N$" @default.
- W4285069811 doi "https://doi.org/10.48550/arxiv.2207.04460" @default.
- W4285069811 hasPublicationYear "2022" @default.
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