Radial regular and rupture solutions for a PDE problem with gradient term and two parameters

Abstract

We investigate radial solutions for the problem \[ { − Δ U = λ + δ | ∇ U | 2 1 − U , U > 0 a m p ; in B , U = 0 a m p ; on ∂ B , \begin {cases} \displaystyle -\Delta U=\frac {\lambda +\delta |\nabla U|^2}{1-U},\; U>0 & \text {in}\ B,\\ U=0 & \text {on}\ \partial B, \end {cases} \] where B ⊂ R N B\subset \mathbb {R}^N ( N ≥ 2 ) (N\geq 2) denotes the open unit ball and λ , δ > 0 \lambda , \delta >0 are real numbers. Two classes of solutions are considered in this work: (i) regular solutions, which satisfy 0 > U > 1 0>U>1 in B B , and (ii) rupture solutions, which satisfy U ( 0 ) = 1 U(0)=1 , and thus make the equation singular at the origin. Bifurcation with respect to parameter λ > 0 \lambda >0 is also discussed.

Radially symmetric stationary solutions for a MEMS type reaction–diffusion equation with fringing field

“…The main results of this paper are shown in section 2. The relations between our results and that of [7] are discussed in section 7 as well as Remarks in section 2.…”

“…Additionally, we note that the results follow from the analysis of the dynamics at infinity. Not only these results but also studies in [7] have motivated us to study the multidimensional version of (2). Indeed, Ghergu-Miyamoto [7] considered the following problem…”

“…Not only these results but also studies in [7] have motivated us to study the multidimensional version of (2). Indeed, Ghergu-Miyamoto [7] considered the following problem…”

“…The singularity at the origin and the bifurcation structure of the radial regular/rupture solution were discussed. According to [7], regular solution is to satisfy 0 < U < 1 in B and rupture solution is to satisfy U(0) = 1. They showed a clear difference from the result in [21], considered the case separation by the value taken by δ (0 < δ < 1, δ = 1, and δ > 1), discussed the solution structure.…”

“…with the boundary conditions u(1) = u (0) = 0. They discuss the uniqueness properties of the positive solutions for (4) (cf [7]). The relation between ( 4) and ( 3) is shown in the next subsection.…”