One-dimensional prescribed mean curvature equation with exponential nonlinearity

Steady state solutions of a bi-stable quasi-linear equation with saturating flux

“…Note that the same is true also of the semilinear case ψ(s) = s which gives the Allen-Cahn equation. Pan [5] however studied a variant of the Liouville, Bratu-Gelfand problem, taking an exponential nonlinearity, f (u) = e u , and both in his case and our case of f (u) = u − u 3 , the nonlinearities are non-homogeneous, so that different bifurcation behaviour in λ is in principle possible for different values of L. This is indeed the case as we shall demonstrate below.…”

“…This boundary value problem, for different choices of the nonlinearity f (u) and boundary conditions has received attention from a variety of authors including Pan [5], Bonheure et al [6] and Habets and Omari [7]. In [7], Habets and Omari study (1.3) with Dirichlet boundary conditions, taking f (u) = (u p ) + , for p > 0, and they investigate the influence of the concavity of this choice of f (u) on the multiplicity of solutions to the problem.…”

Sub- and supersolution methods for prescribed mean curvature equations with Dirichlet boundary conditions

“…Exact numbers of classical and non-classical solutions for (4.5) in one-dimensional case have been obtained in [22,25]. The corresponding semilinear problem also has been well investigated, see Joseph and Lundgren [13], Crandall and Rabinowitz [7], Cabré [5] and the references therein.…”

“…The one-dimensional case of (5.8) has been well studied in [22]: there exists a finite number λ * > 0 such that if λ ∈ (0,λ * ], it admits at least one solutionv λ ∈ C 2 [0, R]; if λ >λ * , it has no solution. Thus by (5.7),v λ is a supersolution of (5.8) for every λ ∈ (0,λ * ].…”

Existence of Solutions and Nonnegative Solutions for Prescribed Variable Exponent Mean Curvature Impulsive System Initialized Boundary Value Problems

“…In 7, 8 , the authors discussed the existence of solutions of p r -Laplacian system impulsive boundary value problems. Recently, the existence and asymptotic behavior of solutions of curvature equations have been studied extensively see [9][10][11][12][13][14][15] . In 16 , the authors generalized the usual mean curvature systems to variable exponent mean curvature systems.…”