The Gelfand Problem for the Infinity Laplacian

We study the asymptotic behavior as p → ∞ of the Gelfand problem −Δpu = λeu in Ω ⊂ Rn, u = 0 on ∂Ω. Under an appropriate rescaling on u and λ, we prove uniform convergence of solutions of the Gelfand problem to solutions of min{|∇u|−Λeu, −Δ∞u} = 0 in Ω, u = 0 on ∂Ω. We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of Λ

Analysis of classes of singular steady state reaction diffusion equations

We study positive radial solutions to classes of steady state reaction diffusion problems on the exterior of a ball with both Dirichlet and nonlinear boundary conditions. We study both Laplacian as well as p-Laplacian problems with reaction terms that are p-sublinear at infinity. We consider both positone and semipositone reaction terms and establish existence, multiplicity and uniqueness results. Our existence and multiplicity results are achieved by a method of sub-supersolutions and uniqueness results via a combination of maximum principles, comparison principles, energy arguments and a-priori estimates. Our results significantly enhance the literature on p-sublinear positone and semipositone problems. Finally, we provide exact bifurcation curves for several one-dimensional problems. In the autonomous case, we extend and analyze a quadrature method, and in the nonautonomous case, we employ shooting methods. We use numerical solvers in Mathematica to generate the bifurcation curves

Bifurcation and multiplicity results for classes of p,qp,q-Laplacian systems

We study positive solutions to boundary value problems of the form \begin{equation*} \begin{cases} -\Delta_{p} u = \lambda \{u^{p-1-\alpha}+f(v)\} & \mbox{in } \Omega,\\ -\Delta_{q} v = \lambda \{v^{q-1-\beta}+g(u)\} & \mbox{in } \Omega,\\ u = 0=v & \mbox{on }\partial\Omega, \end{cases} \end{equation*} where \Delta_{m}u:=\mbox{div}(|\nabla u|^{m-2}\nabla u), m> 1, is the $m$-Laplacian operator of $u$, \lambda> 0, p,q> 1, $\alpha\in(0,p-1)$, $\beta\in(0,q-1)$ and $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $N\geq 1$, with smooth boundary $\partial \Omega$. Here $f,g\colon [0,\infty)\rightarrow \mathbb{R}$ are nondecreasing continuous functions with $f(0)=0=g(0)$. We first establish that for $\lambda\approx 0$ there exist positive solutions bifurcating from the trivial branch $(\lambda,u\equiv 0,v\equiv 0)$ at $(0,0,0)$. We further discuss an existence result for all \lambda > 0 and a multiplicity result for a certain range of $\lambda$ under additional assumptions on $f$ and $g$. We employ the method of sub-super solutions to establish our results

Bifurcation curves for singular and nonsingular problems with nonlinear boundary conditions

We discuss a quadrature method for generating bifurcation curves of positive solutions to some autonomous boundary value problems with nonlinear boundary conditions. We consider various nonlinearities, including positone and semipositone problems in both singular and nonsingular cases. After analyzing the method in these cases, we provide an algorithm for the numerical generation of bifurcation curves and show its application to selected problems