Positive solutions of superlinear indefinite prescribed mean curvature problems

This paper analyzes the superlinear indefinite prescribed mean curvature problem [ -mathrm{div}left({ abla u}/{sqrt{1+| abla u|^2}} ight)=lambda a(x)h(u) quad ext{in }Omega,qquad u=0 quad ext{on } partialOmega, ] where $Omega$ is a bounded domain in $mathbb{R}^N$ with a regular boundary $partial Omega$, $hin C^0(mathbb{R}) $ satisfies $h(s) sim s^{p}$, as $s o0^+$, $p>1$ being an exponent with $p< rac{N+2}{N-2}$ if $Ngeq 3$, $lambda> 0$ represents a parameter, and $ain C^0(overline Omega) $ is a sign-changing function. The main result establishes the existence of positive regular solutions when $lambda$ is sufficiently large, providing as well some information on the structure of the solution set. The existence of positive bounded variation solutions for $lambda$ small is further discussed assuming that $h$ satisfies $h(s) sim s^{q}$ as $s o +infty$, $q>0$ being such that $q< rac{1}{N-1}$ if $Ngeq 2$; thus, in dimension $Nge 2$, the function $h$ is not superlinear at $+infty$, although its potential $H(s) = int_0^sh(t) mathrm{d}t$ is. Imposing such different degrees of homogeneity of $h$ at $0$ and at $+infty$ is dictated by the specific features of the mean curvature operator.