R. Shivaji

Diffusive logistic equation with constant yield harvesting, I: Steady States

We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.

Nonnegative solutions for a class of radially symmetric nonpositone problems

We consider the existence of radially symmetric non-negative solu- tions for the boundary value problem -Au(x) = lf{u(x)) IMI 2) u(x) = 0 ||*|| = 1 where X > 0, f(0) 0 and / is superlinear. We establish existence of non-negative solutions for A small which extends some work of our previous paper on non-positone problems, where we considered the case N = \ . Our work also proves a recent conjecture by Joel Smoller and

Existence Results for Classes of Sublinear Semipositone Problems

We consider the semipositone problem % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

p

{\matrix {-\Delta u (x)= \lambda f (u(x))\ \ \; \ \ \ \ \ x \in \Omega \cr \qquad \qquad \qquad u(x)=0 \ \ \ \;\ \ \ \ x \in \partial \Omega \cr}}

Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity

where λ > 0 is a constant, Ω is a bounded region in Rn with a smooth boundary, and f is a smooth function such that f ′(u) is bounded below, f (0) < 0 and % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

Uniqueness and stability of nonnegative solutions for semipositone problems in a ball

{\rm lim}_{u \rightarrow}+\infty {f(u)\over u}=0.

Diffusion problems with a mixed nonlinear boundary condition

We prove under some additional conditions the existence of a positive solution (1) for λ ∈ I where I is an interval close to the smallest eigenvalue of —Δ with Dirichlet boundary condition and (2) for λ large. We also prove that our solution u for λ large is such that∥u∥ ≔ supx∈Ω ¦u(x)¦ → ∞ as A → ∞. Our methods are based on sub and super solutions. In particular, we use an anti maximum principle to obtain a subsolution for our existence result for λ ∈ I.

Positive solution curves of semipositone problems with concave nonlinearities

We consider the positive solutions of a quasilinear elliptic equation with p-Laplacian, logistic-type growth rate function, and a constant yield harvesting. We use sub-super-solution methods to prove the existence of a maximal positive solution when the harvesting rate is under a certain positive constant.