Existence of ground states and free boundary problems for quasilinear elliptic operators

Abstract

We prove the existence of non-negative non-trivial solutions of the quasilinear equation $\Delta_mu+f(u)=0$ in $\mathbb R^n$ and of its associated free boundary problem, where $\Delta_m$ denotes the $m$-Laplace operator. The nonlinearity $f(u)$, defined for $u>0$, is required to be Lipschitz continuous on $(0,\infty)$, and in $L^1$ on $(0,1)$ with $\int_0^uf(s)\ ds <0$ for small $u>0$; the usual condition $f(0)=0$ is thus completely removed. When $n>m$, existence is established essentially for all subcritical behavior of $f$ as $u\to\infty$, and, with some further restrictions, even for critical and supercritical behavior. When $n=m$ we treat various exponential growth conditions for $f$ as $u\to\infty$, while when $n <m$ no growth conditions of any kind are required for $f$. The proof of the main results moreover yield as a byproduct an a priori estimate for the supremum of a ground state in terms of $n$, $m$ and elementary parameters of the nonlinearity. Our results are thus new and unexpected even for the semilinear equation $\Delta u+f(u)=0$. The proofs use only straightforward and simple techniques from the theory of ordinary differential equations; unlike well known earlier demonstrations of the existence of ground states for the semilinear case, we rely neither on critical point theory [6] nor on the Emden-Fowler inversion technique [2, 3].