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

By Filippo Gazzola, J. Serrin, Moxun Tang. Advances in Differential Equations, 2000

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We prove the existence of non-negative non-trivial solutions of the quasilinear equation ¢mu + f(u )=0i nR n and of its associated free boundary problem, where ¢m denotes the m-Laplace operator. The nonlinearity f(u), de- flned for u> 0, is required to be Lipschitz continuous on (0;1), and in L 1 on (0; 1) with R u 0 f(s) ds 0; the usual condition f(0) = 0 is thus com- pletely removed. When n>m , existence is established essentially for all subcritical behavior of f as u!1, and, with some further restrictions, even for critical and supercritical behavior. When n = m we treat various exponential growth condi- tions for f as u !1 , 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 ¢u +f(u )=0 . The proofs use only straightforward and simple techniques from the theory of ordinary difierential 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).

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/20595cb545cd1bd8613756483c83438f35a053e0

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Citations in the corpus

Existence, non-existence and regularity of radial ground states for p-Laplacian equations with singular weights (2008, background): “[13,19,22] for the no weighted case, that is when β = 0 in (1.”