Maya Chhetri

Analysis of positive solutions for classes of quasilinear singular problems on exterior domains

Abstract We consider the problem { - Δ p ⁢ u = K ⁢ ( x ) ⁢ f ⁢ ( u ) u δ in ⁢ Ω e , u ⁢ ( x ) = 0 on ⁢ ∂ ⁡ Ω , u ⁢ ( x ) → 0 as ⁢ | x | → ∞ , \left\{\begin{aligned} \displaystyle{-}\Delta_{p}u&\displaystyle=K(x)\frac{f(u% )}{u^{\delta}}&&\displaystyle\text{in }\Omega^{e},\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\\ \displaystyle u(x)&\displaystyle\to 0&&\displaystyle\text{as }|x|\to\infty,% \end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N > 2 {N>2} ) is a simply connected bounded domain containing the origin with C 2 {C^{2}} boundary ∂ ⁡ Ω {\partial\Omega} , Ω e := ℝ N ∖ Ω ¯ {\Omega^{e}:=\mathbb{R}^{N}\setminus\overline{\Omega}} is the exterior domain, 1 < p < N {1<p<N} and 0 ≤ δ < 1 {0\leq\delta<1} . In particular, under an appropriate decay assumption on the weight function K at infinity and a growth restriction on the nonlinearity f, we establish the existence of a positive weak solution u ∈ C 1 ⁢ ( Ω e ¯ ) {u\in C^{1}(\overline{\Omega^{e}})} with u = 0 {u=0} pointwise on ∂ ⁡ Ω {\partial\Omega} . Further, under an additional assumption on f, we conclude that our solution is unique. Consequently, when Ω is a ball in ℝ N {\mathbb{R}^{N}} , for certain classes of K ⁢ ( x ) = K ⁢ ( | x | ) {K(x)=K(|x|)} , we observe that our solution must also be radial.

Existence of positive solutions for a class of p -Laplacian superlinear semipositone problems

We consider a quasilinear elliptic problem of the form where λ > 0 is a parameter, 1 p Ω is a strictly convex bounded domain in ℝ N , N > p , with C 2 boundary ∂Ω . The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone ( f (0) p -superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L ∞ a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

Collaborative Mathematics and Statistics Research

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A Game Theoretical Analysis of the Mating Sign Behavior in the Honey Bee

Queens of the honey bee, Apis mellifera (L.), exhibit extreme polyandry, mating with up to 45 different males (drones). This increases the genetic diversity of their colonies, and consequently their fitness. After copulation, drones leave a mating sign in the genital opening of the queen which has been shown to promote additional mating of the queen. On one hand, this signing behavior is beneficial for the drone because it increases the genetic diversity of the resulting colony that is to perpetuate his genes. On the other hand, it decreases the proportion of the drone’s personal offspring among colony members which is reducing drone fitness. We analyze the adaptiveness and evolutionary stability of this drone’s behavior with a game-theoretical model. We find that theoretically both the strategy of leaving a mating sign and the strategy of not leaving a mating sign can be evolutionary stable, depending on natural parameters. However, the signing strategy is not favored for most scenarios, including the cases that are biologically plausible in reference to empirical data. We conclude that leaving a sign is not in the interest of the drone unless it serves biological functions other than increasing subsequent queen mating chances. Nevertheless, our analysis can also explain the prevalence of such a behavior of honey bee drones by a very low evolutionary pressure for an invasion of the nonsigning strategy.

On the existence of multiple positive solutions to some superlinear systems

We use the method of upper and lower solutions combined with degree-theoretic techniques to prove the existence of multiple positive solutions to some superlinear elliptic systems of the form on a smooth, bounded domain Ω⊂ℝ n with Dirichlet boundary conditions, under suitable conditions on g 1 and g 2 . Our techniques apply generally to subcritical, superlinear problems with a certain concave–convex shape to their nonlinearity.

Superlinear elliptic systems with reaction terms involving product of powers

Abstract We consider a system of the form − Δ u = λ g 1 ( x , u , v ) in Ω ; − Δ v = λ g 2 ( x , u , v ) in Ω ; u = 0 = v on ∂ Ω , where λ > 0 is a parameter, Ω ⊂ R N ( N ≥ 2 ) is a bounded domain with sufficiently smooth boundary ∂ Ω (a bounded open interval if N = 1 ). Here g i ( x , s , t ) : Ω × [ 0 , + ∞ ) × [ 0 , + ∞ ) → R ( i = 1 , 2 ) are Caratheodory functions that exhibit superlinear growth at infinity involving product of powers of u and v . Using re-scaling argument combined with Leray–Schauder degree theory and a version of Leray–Schauder continuation theorem, we show that the system has a connected set of positive solutions for λ small.

Proving the “Proof”: Interdisciplinary Undergraduate Research Positively Impacts Students

The math biology program at UNCG has been running since 2006 when we first received the funding from NSF. Every year, we provided integrated research projects at the interface of biology and mathematics to eight UNCG undergraduate students who worked in interdisciplinary teams. Up to date, our project resulted in 32 peer-reviewed publications and over 200 presentations; this demonstrates the extent to which undergraduate research can produce genuine scientific advancement. Moreover, our program also prepared UNCG students for rigorous interdisciplinary graduate studies and career opportunities and set them on a path toward productive careers as twenty-one century scientists and educators. We hope our experience will motivate and encourage others to pursue similar efforts.

Revisiting the variance-based selection model of diploid drone production for multiple mating in honey bees

Abstract The high number of mates of honeybee queens has lead to the proposal of several adaptive explanations. The competing hypotheses to explain multiple mating in honeybees and some other social insects have been mostly evaluated empirically with comprehensive theoretical analysis lacking behind. We report on the mathematical analysis of the diploid drone hypothesis for multiple mating, which suggests that multiple mating evolved as a safeguard against the production of infertile male offspring. In accordance with earlier models, our analysis shows that multiple mating does not reduce the average value of diploid drone production but reduces its variance. We combine this observation with a colony growth model to assess the impact of this reduction in variance to the colony fitness. Considering a plausible parameter space for the honeybee, we conclude that the reduction in variance of diploid drone production can be a significant selective force for multiple mating. We have also described rules of a game for which a problem of finding the best strategy is equivalent to the above biological problem of bee mating. We made a significant progress in the general solution of this game and conjectured that the best strategy is strongly related to the geometry of rational numbers.

Existence of a positive solution for a -Laplacian semipositone problem

We consider the boundary value problem in satisfying on, where on, is a parameter, is a bounded domain in with boundary, and for. Here, is a nondecreasing function for some satisfying (semipositone). We establish a range of for which the above problem has a positive solution when satisfies certain additional conditions. We employ the method of subsuper solutions to obtain the result.