Existence of quasilinear elliptic equations with prescribed limiting behavior

Abstract

We consider quasilinear elliptic equations Δ p u + f ( u ) = 0 in the quarter-plane Ω, with zero Dirichlet data. For some general nonlinearities f, we prove the existence of a positive solution with a prescribed limiting profile. The question is motivated by the result in (Adv. Nonlinear Stud. 13(1) (2013) 115–136), where the authors establish that for solutions u ( x 1 , x 2 ) of the preceding Dirichlet problem, lim x 1 → ∞ u ( x 1 , x 2 ) = V ( x 2 ), where V is a solution of the corresponding one-dimensional problem with V ( + ∞ ) = z and z is a root of f. Starting with such a profile V and a carefully selected z, the authors of this paper apply Perron’s method in order to prove the existence of a solution u with limiting profile V. The work in this paper is similar in spirit to that in (Math. Methods Appl. Sci. 39(14) (2016) 4129–4138), where the authors compare the sub and the super solutions by using arguments based on the strong maximum principle for semilinear equations. However, for the quasilinear case, such a maximum principle is lacking. This difficulty is overcome by employing a less classical weak sweeping principle that requires a careful boundary analysis.