Existence and multiplicity of positive solutions for one-dimensional prescribed mean curvature equations
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RESEARCH
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Existence and multiplicity of positive solutions for one-dimensional prescribed mean curvature equations Yanqiong Lu, Ruyun Ma* and Hongliang Gao Dedicated to Professor Ivan T. Kiguradze for his merits in the theory of differential equations *
Correspondence: [email protected] Department of Mathematics, Northwest Normal University, Lanzhou, 730070, P.R. China
Abstract In this work, we establish the existence and multiplicity results of positive solutions for one-dimensional prescribed mean curvature equations. Our approach is based on fixed point index theory for completely continuous operators which leave invariant a suitable cone in a Banach space of continuous functions. MSC: 34B10; 34B18 Keywords: mean curvature equations; positive solutions; existence; fixed point index
1 Introduction The prescribed mean curvature problems like
– div( √
Du ) = f (x, u), +κ(Du)
u = ,
x ∈ ∂
x ∈ ,
have attracted much attention in recent years, see [–] and the references therein. Since the problem is quasilinear non-uniformly elliptic, it is more difficult to study the existence of classical solutions. The greatest obstacle is the lack of gradient estimate, such kind of estimate does not hold in general and boundary gradient blow-up may occur. This leads to some new phenomena very different from those in semilinear problems. Many well-known results of semilinear problems have to be reconsidered for this quasilinear problem. Motivated by the search for solutions of the above problem, many authors (see [–]) studied the existence of (positive) solutions for one-dimensional prescribed mean curvature equations with Dirichlet boundary conditions
–( √
u ) +κ(u )
= λf (u),
x ∈ (, ),
(.)
u() = u() = , where κ > is a constant, f ∈ C([, ∞), [, ∞)) and f (u) > for u > and x ∈ [, ]. © 2014 Lu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Lu et al. Boundary Value Problems 2014, 2014:120 http://www.boundaryvalueproblems.com/content/2014/1/120
Page 2 of 12
Note that if κ = , problem (.) is degenerate to the second-order ordinary differential equation boundary value problems –u = λf (u), x ∈ (, ), u() = u() = .
(.)
The existence of (positive) solutions of (.) has been well known with various qualitative assumptions of the nonlinearity f , see [, ] and the references therein. If κ = , Bonheure et al. [], Habets and Omari [], Kusahara and Usami [], Pan and Xing [, ] studied the existence of (positive) solutions of (.) by using the variational method, lower and upper solutions method and time mapping method, respectively. However, to the best of our knowledge, the existence and multiplicity of positive solutions for (.) are relatively few by the fixed point index theory. In this paper, based on the
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