Ground States for the Prescribed Mean Curvature Equation: The Supercritical Case

Abstract

We shall consider here the question of existence and nonexistence of ground states for the prescribed mean curvature equation in ℝⁿ (n > 2), that is, we consider solutions of the problem

$$ \left. {\begin{array}{*{20}{c}} {div\left( {\frac{{Du}}{{{{\left( {1 + + {{\left| {Du} \right|}^2}} \right)}^{{1/2}}}}}} \right) + f(u) = 0\;in {\mathbb{R}^n}} \\ {u > 0\quad in\,{\mathbb{R}^n}} \\ {u(x) \to 0\;as\,x \to \infty } \\ \end{array} } \right\} $$

((1.1))

where Du denotes the gradient of u. The function f(u), defined for u > 0, will be assumed throughout to satisfy the following hypotheses:

$$ f \in {C^1}\left[ {0,\infty } \right) $$

((H1))

(H2) f(0) = 0, and there exists a number a ≥ 0 such that

$$ f \in {C^1}\left[ {0,\infty } \right) $$

if a > 0 we require

$$ I(u) = \int\limits_B {F\left( {x,u,Du(x)} \right)} \;dx $$

.