## Existence of ground states and free boundary problems for quasilinear elliptic operators.(English)Zbl 0987.35064

Summary: The authors prove the existence of nonnegative nontrivial 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 $$\int^u_0 f(s)ds <0$$ 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. The 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, the authors rely neither on critical point theory nor on the Emden-Fowler inversion technique.

### MSC:

 35J70 Degenerate elliptic equations 35J60 Nonlinear elliptic equations