In this paper we characterize the existence of principal eigenvalues for a general class of linear weighted second order elliptic boundary value problems subject to a very general class of mixed boundary conditions. Our theory is a substantial extension of the classical theory by P. Hess and T. Kato (1980, Comm. Partial Differential Equations 5, 999-1030. In obtaining our main results we must give a number of new results on the continuous dependence of the principal eigenvalue of a second order linear elliptic boundary value problem with respect to the underlying domain and the boundary condition itself. These auxiliary results complement and in some sense complete the theory of D. Daners and E. N. Dancer (1997, J. Differential Equations 138, 86-132). The main technical tool used throughout this paper is a very recent characterization of the strong maximum principle in terms of the existence of a positive strict supersolution due to H. Amann and J. López-Gómez (1998, J. Differential Equations 146, 336-374).

This paper shows the existence and the uniqueness of the positive solution (t) of the singular boundary value problemwhere f is a continuous non-decreasing function such that f (0) 0, and h is a non-negative function satisfying the Keller-Osserman condition. Moreover, it also ascertains the exact blow-up rate of (t) at t = 0 in the special case when there exist H > 0 and p > 1 such that h(u) ∼ H u p for sufficiently large u. Naturally, the blow-up rate of the problem in such a case equals its blow-up rate for the very special, but important, case when h(u) = H u p for all u 0. So, our results are substantial improvements of some previous findings of [J. López-Gómez, Uniqueness of large solutions for a class of radially symmetric elliptic equations, in: S. Cano-Casanova, J. López-Gómez, C. Mora-Corral (Eds.), Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, World Scientific, 2005, pp. 75-110] and [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385-439].

This paper adapts a technical device going back to [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385-439] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677-686]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180-3203]. Precisely, we show that if Ω is a ball, or an annulus, f ∈ C[0, ∞) is positive and non-decreasing, V ∈ C[0, ∞) ∩ C 2 (0, ∞) satisfies V (0) = 0, V (u) > 0, V (u) 0, for every u > 0, and V (u) ∼ H u p−1 as u ↑ ∞, for some H > 0 and p > 1, then, for each λ 0, −Δu = λu − f dist(x, ∂Ω) V (u)u possesses a unique positive large solution in Ω, L, which must be radially symmetric, by uniqueness, and we can estimate the exact blow-up rate of L(x) at ∂Ω in terms of p, H and f (see Theorem 1.1).

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In this paper we construct an example of superlinear indefinite weighted elliptic mixed boundary value problem exhibiting a mushroom shaped compact component of positive solutions emanating from the trivial solution curve at two simple eigenvalues of a related linear weighted boundary value problem. To perform such construction we have to adapt to our general setting some of the rescaling arguments of H. Amann and J. López-Gómez [2, Section 4] to get a priori bounds for the positive solutions. Then, using the theory of [1], [4] and [5], we give some sufficient conditions on the nonlinearity and the several potentials of our model setting so that the set of values of the parameter for which the problem possesses a positive solution is bounded. Finally, the existence of the component of positive solutions emanating from the trivial curve follows from the unilateral results of P. H. Rabinowitz ([18], [14]). Monotonicity methods, re-scaling arguments, Liouville type theorems, local bifurcation and global continuation are among the main technical tools used to carry out our analysis. 2000 Mathematics Subject Classification. 35B50, 35B32, 35B60, 35B45, 35J65.