Fourth order semipositone problems on R^N
Speaker 
Dr. Nirjan Biswas, Post Doc Fellow, TIFR CAM


When 
Feb 08, 2023
from 04:00 PM to 05:00 PM 
Where  Auditorium,Ground Floor 
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\textbf{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 $C_c^{\infty}(\mathbb{R}^N)$ with respect to $(\int_{\mathbb{R}^N} (\Delta u )^2)^{1/2}$. For $f_a$ satisfying subcritical nonlinearity and a weaker AmbrosettiRabinowitz 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.