Existence results for classes of <inline-formula><graphic file="1687-2770-2006-87483-i1.gif"/></inline-formula>-Laplacian semipositone equations

Abstract

<p/> <p>We study positive <inline-formula><graphic file="1687-2770-2006-87483-i2.gif"/></inline-formula> solutions to classes of boundary value problems of the form <inline-formula><graphic file="1687-2770-2006-87483-i3.gif"/></inline-formula> in <inline-formula><graphic file="1687-2770-2006-87483-i4.gif"/></inline-formula> on <inline-formula><graphic file="1687-2770-2006-87483-i5.gif"/></inline-formula>, where <inline-formula><graphic file="1687-2770-2006-87483-i6.gif"/></inline-formula> denotes the <inline-formula><graphic file="1687-2770-2006-87483-i7.gif"/></inline-formula>-Laplacian operator defined by <inline-formula><graphic file="1687-2770-2006-87483-i8.gif"/></inline-formula>; <inline-formula><graphic file="1687-2770-2006-87483-i9.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i10.gif"/></inline-formula> is a parameter, <inline-formula><graphic file="1687-2770-2006-87483-i11.gif"/></inline-formula> is a bounded domain in <inline-formula><graphic file="1687-2770-2006-87483-i12.gif"/></inline-formula>; <inline-formula><graphic file="1687-2770-2006-87483-i13.gif"/></inline-formula> with <inline-formula><graphic file="1687-2770-2006-87483-i14.gif"/></inline-formula> of class <inline-formula><graphic file="1687-2770-2006-87483-i15.gif"/></inline-formula> and connected (if <inline-formula><graphic file="1687-2770-2006-87483-i16.gif"/></inline-formula>, we assume that <inline-formula><graphic file="1687-2770-2006-87483-i17.gif"/></inline-formula> is a bounded open interval), and <inline-formula><graphic file="1687-2770-2006-87483-i18.gif"/></inline-formula> for some <inline-formula><graphic file="1687-2770-2006-87483-i19.gif"/></inline-formula> (semipositone problems). In particular, we first study the case when <inline-formula><graphic file="1687-2770-2006-87483-i20.gif"/></inline-formula> where <inline-formula><graphic file="1687-2770-2006-87483-i21.gif"/></inline-formula> is a parameter and <inline-formula><graphic file="1687-2770-2006-87483-i22.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i23.gif"/></inline-formula> function such that <inline-formula><graphic file="1687-2770-2006-87483-i24.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i25.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i26.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i27.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i28.gif"/></inline-formula>. We establish positive constants <inline-formula><graphic file="1687-2770-2006-87483-i29.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i30.gif"/></inline-formula> such that the above equation has a positive solution when <inline-formula><graphic file="1687-2770-2006-87483-i31.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i32.gif"/></inline-formula>. Next we study the case when <inline-formula><graphic file="1687-2770-2006-87483-i33.gif"/></inline-formula> (logistic equation with constant yield harvesting) where <inline-formula><graphic file="1687-2770-2006-87483-i34.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2006-87483-i35.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i36.gif"/></inline-formula> function that is allowed to be negative near the boundary of <inline-formula><graphic file="1687-2770-2006-87483-i37.gif"/></inline-formula>. Here <inline-formula><graphic file="1687-2770-2006-87483-i38.gif"/></inline-formula> is a <inline-formula><graphic file="1687-2770-2006-87483-i39.gif"/></inline-formula> function satisfying <inline-formula><graphic file="1687-2770-2006-87483-i40.gif"/></inline-formula> for <inline-formula><graphic file="1687-2770-2006-87483-i41.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2006-87483-i42.gif"/></inline-formula>, and <inline-formula><graphic file="1687-2770-2006-87483-i43.gif"/></inline-formula>. We establish a positive constant <inline-formula><graphic file="1687-2770-2006-87483-i44.gif"/></inline-formula> such that the above equation has a positive solution when <inline-formula><graphic file="1687-2770-2006-87483-i45.gif"/></inline-formula> Our proofs are based on subsuper solution techniques.</p