Existence and Exact Multiplicity for Quasilinear Elliptic Equations in Quarter-Spaces

Abstract

<p>We consider positive solutions of quasilinear elliptic problems of the form Δ<i>p^u</i> + ƒ(<i>u</i>) = 0 over the quarter-space <i>Q</i> = {<i>x</i> ∈ ℝ<i>^N</i> : <i>x</i>₁ > 0, <i>x</i>₂ > 0}, with <i>u</i> = 0 on ∂<i>Q</i>. For a general class of nonlinearities ƒ ≥ 0 with finitely many positive zeros, we show that, for each <i>z</i> > 0 such that ƒ(<i>z</i>) = 0, there is a bounded positive solution satisfying </p> <p>lim u(<i>x</i>1, <i>x</i>2, ..., <i>x_N</i>) = <i>V</i> (x2), <br/> <i>x</i>1→∞ <br/>lim u(<i>x</i>1, <i>x</i>2, ..., <i>x_N</i>) = <i>V</i> (<i>x</i>1),<br/><i>x</i>2→∞</p> <p>where <i>V</i> is the unique solution of the one-dimensional problem</p> <p>ΔpV + ƒ(<i>V</i>) = 0 in [0,∞), <i>V</i> (0) = 0, <i>V</i> (<i>t</i>) > 0 for <i>t</i> > 0, <i>V</i> (∞) = <i>z</i>.</p> <p>When <i>p</i> = 2, we show further that such a solution is unique, and there are no other types of bounded positive solutions to the quarter-space problem. Thus in this case the number of bounded positive solutions to the quarter-space problem is exactly the number of positive zeros of ƒ.</p>