[...]... B(-V(z)) and 2) are the boundary operators defined by (3) Proof Let U A be a positive solution of Problem ( 1 ) ~ Then, thanks to Eq ( 8 ) , and owing to the monotonicity of the principal eigenvalue with respect to the domain (cf Proposition 3.2 of S Cano-Casanova and J L6pez-G6mez5) and the dominance of the principal eigenvalue under Dirichlet boundary conditions (cf Proposition 3.1 of S Cano-Casanova and. .. z ) b(z)u;-')] + < o:'[L(X), B(-V(z) b(z)u;-', a:)] I c:'[L(X), 27 + On the other hand, thanks to the monotonicity of the principal eigenvalue with respect to the potential (cf Proposition 3.3 of S Cano-Casanova and J L6pez-G6mez5) and with respect to the weight on the boundary (cf Proposition 3.5 of S Cano-Casanova and J L6pez-G6mez5),it follows from Eq (8) that + 0 = @[L(X) a(z>u;-l, B(-V(z) + b(z)uI-')]... Theorem 1.1 are satisfied Acknowledgments I am delighted to thank my colleague and friend J Lbpez-Gbmez, for inviting to me to contribute with this paper in this memory in tribute to the great mathematician, colleague and friend, J Esquinas Candenas Also, I want to thank to J Esquinas, the advice about the lectures that I had to take to complete my scientific formation in the postgraduate References 1 H Amann,... L6pez-G6mez, Nonlinear Analysis T.M.A 47, 1797-1808, (2001) 7 S CaneCasanova, Nonlinear Analysis T.M.A., 49,361-430, (2002) 8 S Cano-Casanova and J Lbpez-Gbmez, Electronic Journal of Differential Equations, 74,1-41, (2004) 9 S Cano-Casanova, O n the existence and uniqueness of positive solutions of the Logistic elliptic B VP with nonlinear mixed boundary conditions, Nonlinear Analysis, To appear 10... (1'4 Now, using the boundary operator B ( k ( z ) ) defined by Eq (5), Problem ( 1 1 ) ~ be written in the form can + 0 { B(-V(z) + b(z)G(z))(ul ua) (L(X) a ( z ) F ( z ) ) ( u 1 ~ - 2= ) - in R = 0 on dR (13) Thanks to the monotonicity of the principal eigenvalue with respect to the potential (cf Proposition 3.3 of S Cano-Casanova and J L6pez-G6mez5) and with respect to the weight on the boundary (cf... of type (1) to (3) have been considered in the framework of extrapolation theory, especially in the cases p = 1 and p = 00 We refer to the papers by Jawerth and Milman [19] and Milman [22] Let me give an application of the case (ii) when p = 1 and b = 1 It is taken from the book [14] and refers to the Hardy-Littlewood maximal function If p = 1, then l/p”J = 1 - 1/23’ So 2j l/(p”j - l), and we get that... class C2 with dRonR) > 0, we will denote by B ( k ( z ) ,no) the boundary operator dist (rl, build up from B ( k ( z ) )by and by I$" [C,B ( k ( z ) ,Ro)], the principal eigenvalue of (C, B ( k ( z ) ,Ro), 00) To develop the mathematical analysis of the next sections, are essential the different monotonicity properties of op[C, B ( k ( z ) ) ]joint with the continuous dependence of it with respect to perturbations... elliptic in R with the same ellipticity constant as the operator L The following theorem collects the main results of this work Theorem 1.1 Problem ( 1 ) possesses a positive solution, if and only if ~ ': 0 [,c(x>,731 stand for the principal eigenvalue in 0 and s2:, respectively, of L(A), subject to the boundary operators B ( - V ( z ) ) and D,respectively, being where ?[C(X), B ( - V ( ~ ) ) ] and Moreover,... arbitrarily large and ~ bounded away from zero in fi 0 : : Proof In order not to enlarge the exposition, we are going to prove the result in a particular case, sufficiently general, being this proof easily extended to the case when for instance, a(.) belongs to the general class of nonnegative measurable potentials d ( R ) of admissible potentials in R introduced in the works due to S Cano-Casanova and J L6pez-G6mez5,... R;, and lim 6-0+ a; = at, in the sense of Definition 6.1 of S Cano-Casanova and J L6pez-G6mez5 Thus, it follows from Theorem 4.2 of J L6pez-G6mez1O, and Theorem 7.1 of S Cano-Casanova and J L6pez-G6mez5, that 6-Of lim cp' [L(x), ( n , = c [L(x), ( n A ),~ ~ R:)] ? ~ R, (29) and & to+ c [L(A), = 0; [L(x>, lim : : D] : DI (30) Now it follows from Eqs (21), (23), (26), (29), (30), and Proposition 3.2 and . Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology This page intentionally left blankThis page intentionally left blank Spectral. is available from the British Library SPECTRAL THEORY AND NONLINEAR ANALYSIS WITH APPLICATIONS TO SPATIAL ECOLOGY Copyright 0 2005 by World Scientific