Existence and non-existence results for fully nonlinear elliptic systems

Abstract

Abstract. We study systems of two elliptic equations, with right-hand sides with general power-like superlinear growth, and left-hand sides which are of Isaac's or Hamilton-Jacobi-Bellman type (however our results are new even for linear lefthand sides). We show that under appropriate growth conditions such systems have positive solutions in bounded domains, and that all such solutions are bounded in the uniform norm. We also get nonexistence results in unbounded domains.

Stationary States of Reaction-Diffusion and Schrödinger Systems with Inhomogeneous or Controlled Diffusion

“…The only difference is that now the modified functionsũ j ,ṽ j satisfy a system where appears the elliptic operator F (D 2ũ j , λ j Dũ j , x j + λ j y), and λ j → 0. By using the global C 1,α -estimates for Isaacs operators and Ascoli's theorem, we can extract a subsequence of (ũ j ,ṽ j ) which converges in C 1 loc (the way to perform such a limit argument for fully nonlinear operators is described in extenso in [32]). After the passage to the limit by Lemma 3.8, we obtain a bounded nonnegative viscosity solution (U, V ) of…”

“…As far as the latter case is concerned, it is proved in [17] and Theorem 3.1 in [31] that the nonexistence of solutions of the equation −F (D 2 u) = f (u) in R n (and even in R n−1 ) implies the nonexistence of positive solutions in a half-space of R n , for every rotationally invariant operator F and locally Lipschitz nonlinearity f (in [31] only Pucci operators were considered, but the proof is the same for every rotationally invariant operator). An extension of this result to systems is proved in [32]. Even stronger results are known if the elliptic operator is the Laplacian, [13].…”

“…We observe that existence and non-existence for fully nonlinear systems with a different class of nonlinearities (with cooperative and fully coupled leading terms such as Lane-Emden type systems, see (20) below) can be found in [32]. We refer to that paper, as well to [9] and [23], for more examples and references on problems where fully nonlinear systems appear.…”

“…Note that results on classification of bounded solutions of fully nonlinear systems in cones were previously obtained in [32], for a different type of nonlinearities (of Lane-Emden type), and with different proofs. On the other hand, the statement (b) above is to our knowledge the first classification result for unbounded solutions of systems with operators in nondivergence form.…”

Some estimates and maximum principles for weakly coupled systems of elliptic PDE

“…where we have used (10), (11), and have noted f i,ε = f + i + ε j c ij . In the sequel we shall in general denote with f i,ε any function which converges to f…”