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.


35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations