Search

This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [ABSTRACT FROM AUTHOR]

The aim of this paper is analyzing existence, multiplicity, and regularity issues for the positive solutions of the quasilinear Neumann problem { − (u ′ / 1 + (u ′) 2 ) ′ = λ a (x) f (u) , 0 < x < 1 , u ′ (0) = u ′ (1) = 0. Here, (u ′ / 1 + (u ′) 2 ) ′ is the one-dimensional curvature operator, λ ∈ R is a parameter, which is generally taken positive, the weight a (x) changes sign, and, in most occasions, the function f (u) has a sublinear potential F (u) at ∞. Our discussion displays the manifold patterns occurring for these solutions, depending on the behavior of the potential F (u) at u = 0 , and, possibly, at infinity, and of the weight function a (x) at its nodal points. [ABSTRACT FROM AUTHOR]

This paper deals with the semilinear elliptic problem - Δ u = λ m (x) u - [ a (x) + ε b (x) ] u p in Ω , B u = 0 on ∂ Ω , where p > 1 , λ > 0 , m , a , b ∈ C (Ω ¯) , with a ⪈ 0 , b ⪈ 0 , Ω is a bounded C 2 domain of R N ( N ≥ 1 ), B is a general classical mixed boundary operator, and ε ≥ 0 . Thus, a(x) and b(x) can vanish on some subdomain of Ω and the weight function m(x) can change sign in Ω . Through this paper we are always considering classical solutions. First, we characterize the existence of positive solutions of this problem in the special case when ε = 0 . Then, we investigate the sharp patterns of the positive solutions when ε ↓ 0 and ε ↑ ∞ . Our study reveals how the existence of sharp profiles is determined by the behavior of b(x). [ABSTRACT FROM AUTHOR]

We give a series of very general sufficient conditions in order to ensure the uniqueness of positive large solutions for - Δ u + f (x , u) = 0 in a bounded domain Ω where f : Ω ¯ × R ↦ R + is a continuous function, such that f (x , 0) = 0 for x ∈ Ω ¯ , and f (x , r) > 0 for x in a neighborhood of ∂ Ω and all r > 0 . [ABSTRACT FROM AUTHOR]

The aim of this paper is characterizing the development of singularities by the positive solutions of the quasilinear indefinite Neumann problem − (u ′ / 1 + (u ′) 2 ) ′ = λ a (x) f (u) in (0 , 1) , u ′ (0) = 0 , u ′ (1) = 0 , where λ ∈ R is a parameter, a ∈ L ∞ (0 , 1) changes sign once in (0 , 1) at the point z ∈ (0 , 1) , and f ∈ C (R) ∩ C 1 [ 0 , + ∞) is positive and increasing in (0 , + ∞) with a potential, ∫ 0 s f (t) d t , superlinear at +∞. In this paper, by providing a precise description of the asymptotic profile of the derivatives of the solutions of the problem as λ → 0 + , we can characterize the existence of singular bounded variation solutions of the problem in terms of the integrability of this limiting profile, which is in turn equivalent to the condition (∫ x z a (t) d t) − 1 2 ∈ L 1 (0 , z) and (∫ x z a (t) d t) − 1 2 ∈ L 1 (z , 1). No previous result of this nature is known in the context of the theory of superlinear indefinite problems. [ABSTRACT FROM AUTHOR]

Abstract In this paper we construct a general class of non-monotone positive nonlinearities, f (u) , for which the differential equation u ′ ′ = f (u) has a unique large solution in (− T , T). In most of the available literature it is required f (u) to be strictly increasing in order to establish the uniqueness of the large solution. [ABSTRACT FROM AUTHOR]

In this paper, we ascertain the global λ -structure of the set of positive and negative solutions bifurcating from u = 0 for the semilinear elliptic BVP − d Δ u = λ 〈 a , ∇ u 〉 + u + λ u 2 − u q in Ω , u = 0 on ∂ Ω , according to the values of d > 0 and the integer number q ≥ 4. Figs. 1.1–1.3 summarize the main findings of this paper according to the values of d and q. Note that the role played by the parameter λ in this model is very special, because, besides measuring the strength of the convection, it quantifies the amplitude of the nonlinear term λ u 2 . We regard to this problem as a mathematical toy to generate solution loops and isolas in Reaction Diffusion equations. [ABSTRACT FROM AUTHOR]

This paper discusses the Keller–Osserman condition from a dynamical perspective in order to obtain a rather astonishing multiplicity result of large solutions. It turns out that, for any given increasing positive function f ( u ) that satisfies the Keller–Osserman condition, destroying the monotonicity of f ( u ) on a compact set with arbitrarily small measure can originate an arbitrarily large number of explosive solutions. Moreover, some counterexamples to an important result of [6] are given. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of the spatially-heterogeneous diffusive Lotka–Volterra competing species model. It focuses special attention in ascertaining the linear stability and multiplicity of the coexistence steady states. One of our main findings establishes that, as soon as any steady-state solution of the non-spatial model is linearly unstable somewhere in the inhabiting territory, Ω , any steady state of the spatial counterpart perturbing from it therein (as the diffusion rates, d 1 , d 2 , move away from 0 ) must be linearly unstable. From this general principle one can derive a number of rather astonishing consequences, as the multiplicity of the coexistence steady states when the non-spatial model exhibits founder control competition somewhere in Ω , say Ω bi , even if Ω bi is negligible empirically. Actually, this is the first available multiplicity result for small diffusion rates. Finally, based on a celebrated identity by M. Picone (1910), we are able to establish a new, rather striking, uniqueness result valid for general spatially heterogeneous models. [ABSTRACT FROM AUTHOR]

We establish the uniqueness of the positive solution of the singular problem (1.1) through some standard comparison techniques involving the maximum principle. Our proofs do not invoke to the blow-up rates of the solutions, as in most of the specialized literature. We give two different types of results according to the geometrical properties of $$\Omega $$ and the regularity of $$\partial \Omega $$ . Even in the autonomous case, our theorems are extremely sharp extensions of all existing results. Precisely, when $$\mathfrak {a}(x)\equiv 1$$ , it is shown that the monotonicity and superadditivity of f( u) with constant $$C\ge 0$$ entail the uniqueness; f is said to be superadditive with constant $$C\ge 0$$ if This condition, introduced by Marcus and Véron (J Evol Equ 3:637-652, 2004), weakens all previous sufficient conditions for uniqueness, as it will become apparent in this paper. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of the diffusive Lotka-Volterra competition model under small dispersion rates and Dirichlet boundary conditions. Its main goal is ascertaining the connections between the qualitative behavior of the positive solutions of the parabolic model for small diffusions and the dynamics of its associated non-spatial model given by switching them off. After sharpening very substantially some previous results of Furter and López-Gómez (Proc R Soc Edinb 127A:281-336, 1997), we characterizes the singular limit, as diffusions go to zero, of any sequence of coexistence steady-state solutions. It turns out that they must approximate, point-wise in the inhabiting territory, the global attractor of the non-spatial model, uniformly on compact subsets of the habitat zones where a global hyperbolic attractor exist. As a very special consequence of our general theorem, the Dirichlet counterpart of the singular perturbation theorem of Hutson et al. (World Sci Ser Appl Anal 4:501-533, 1995) holds. Further, a multiplicity result is given when the underlying non-spatial model exhibits a founder control competition somewhere in the territory. This is the first multiplicity theorem available in the literature for small diffusivities. [ABSTRACT FROM AUTHOR]

We study the structure of the set of the positive regular solutions of the one-dimensional quasilinear Neumann problem involving the curvature operator − ( u ′ / 1 + ( u ′ ) 2 ) ′ = λ a ( x ) f ( u ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) = 0 . Here λ ∈ R is a parameter, a ∈ L 1 ( 0 , 1 ) changes sign, and f ∈ C ( R ) . We focus on the case where the slope of f at 0 , f ′ ( 0 ) , is finite and non-zero, and the potential of f is superlinear at infinity, but also the two limiting cases where f ′ ( 0 ) = 0 , or f ′ ( 0 ) = + ∞ , are discussed. We investigate, in some special configurations, the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions. [ABSTRACT FROM AUTHOR]

We prove the existence and the multiplicity of positive solutions of the one-dimensional capillarity-type problem − ( u ′ / 1 + ( u ′ ) 2 ) ′ = a ( x ) f ( u ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) = 0 , where a ∈ L 1 ( 0 , 1 ) changes sign and f : [ 0 , + ∞ ) → [ 0 , + ∞ ) is continuous and has a power-like behavior at the origin and at infinity. Our approach is variational and relies on a regularization procedure that yields bounded variation solutions which are of class W loc 2 , 1 , and hence satisfy the equation pointwise almost everywhere, on each open interval where the weight function a has a constant sign. [ABSTRACT FROM AUTHOR]

This paper ascertains the blow-up rates of each of the components of a singular boundary value problem related to a cooperative system of logistic type, in order to establish the uniqueness of the large solution. Astonishingly, the cooperative coupling does not change the blow-up rates of the uncoupled system provided these blow-up rates are sufficiently close, though it changes exactly one of them, keeping invariant the other, when they are bounded away. This seems to be the first time where this change of behavior has been documented in the literature. [ABSTRACT FROM AUTHOR]

This paper establishes a generalized version of the singular perturbation results given by V. Hutson et al. [10, Theorem 4.1] and X. He and W.M. Ni [6, Theorem 4.2 (iii)] . In particular, it ascertains the limiting profiles of the coexistence states of the classical Lotka–Volterra model for two competing species as the diffusion coefficients approximate zero. They are provided by the global attractors of the underlying non-spatial model whenever they exist. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of the minimal large positive solution for a general class of nonlinear cooperative systems including the simplest prototype of García-Melián et al. (2016). Precisely, based on the existence of a large positive supersolution, we can infer the existence of the minimal large positive solution. Moreover, we also give some sufficient easily computable conditions for the existence of a large positive supersolution. Our results generalize, very substantially, some of the findings of García-Melián et al. (2016) adopting a rather novel methodology. [ABSTRACT FROM AUTHOR]

This paper shows the uniqueness of the large solution of a class of radially symmetric sublinear elliptic systems of cooperative type. This result is based on the strong maximum principle, rather than on the blow-up rates of the large solutions, as it is common in most the available results for single equations. [ABSTRACT FROM AUTHOR]

This paper studies an eigenvalue problem associated with a linear parabolic equation and a coupled ordinary differential equation. The existence and the uniqueness of the principal eigenvalue for this eigenvalue problem is first established. Then, the qualitative dependence of the principal eigenvalue with respect to the several parameters involved in the system is analyzed. Finally, these results are applied to a system in flowing habitats with a hydraulic storage zone and light limitation. [ABSTRACT FROM AUTHOR]

This paper aims at proving the existence and the localization of an unbounded connected set of positive regular solutions (λ , u) of the quasilinear Neumann problem − ( u ′ / 1 + (u ′) 2 ) ′ = λ a (x) f (u) , 0 < x < 1 , u ′ (0) = u ′ (1) = 0 , bifurcating from u = 0 as λ → + ∞. Here, ( u ′ / 1 + (u ′) 2 ) ′ is the one-dimensional curvature operator, λ ∈ R is a parameter, the weight a changes sign, and the function f is superlinear at 0. A novel approach is introduced based on the explicit construction of non-ordered sub and supersolutions. [ABSTRACT FROM AUTHOR]

This paper analyses a canonical class of one-dimensional superlinear indefinite boundary value problems of great interest in population dynamics under non-homogeneous boundary conditions; the main bifurcation parameter in our analysis is the amplitude of the superlinear term. Essentially, it continues the analysis of López-Gómez et al. (López-Gómez, J., Tellini, A. & Zanolin, F. (2014) High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Comm. Pure Appl. Anal. 13(1), 1–73) with empty overlapping, by computing the bifurcation diagrams of positive steady states of the model and by proving analytically a number of significant features, which have been observed from the numerical experiments carried out here. The numerics of this paper, besides being very challenging from the mathematical point of view, are imperative from the point of view of population dynamics, in order to ascertain the dimensions of the unstable manifolds of the multiple equilibria of the problem, which measure their degree of instability. From that point of view, our results establish that under facilitative effects in competitive media, the harsher the environmental conditions, the richer the dynamics of the species, in the sense discussed in Section 1. [ABSTRACT FROM PUBLISHER]

Abstract: This paper studies the Cauchy problem for a parabolic–elliptic system in modeling chemotaxis as well as self-attracting particles. In the critical mass case the fine dynamics of the model is ascertained in terms of the structure of the underlying ω-limit sets. According to the results of this paper, any nonnegative radially symmetric bounded solution either stabilizes to a steady-state as , or oscillates between two steady-states. Moreover, a rather general class of nonnegative initial data, not necessarily radially symmetric, for which the associated solutions exhibit a complex oscillatory behavior is constructed; their ω-limit sets consist of a nontrivial topological continuum of steady-states. Besides the technical difficulties inherent to the lack of compactness of the resolvent operators, one has to add the challenge that the problem is utterly non-local. Consequently, thought the basic ideas on the foundations of this paper might be considered classical, most of the proofs throughout are extremely sophisticated and absolutely new in their full generality. [Copyright &y& Elsevier]

Abstract: This paper proves the uniqueness of the positive linearly stable steady-state for a paradigmatic class of superlinear indefinite parabolic problems arising in population dynamics, under nonhomogeneous Dirichlet conditions on the boundary of the domain. The result is absolutely non-trivial, since examples are known for which the model admits an arbitrarily large number of steady-states. Our proof is based on some local and global continuation techniques. Optimal existence and multiplicity results are also obtained through some additional monotonicity and topological techniques. [Copyright &y& Elsevier]

Abstract: We study the qualitative effects of the transport terms on the maximum principle for a family of one-dimensional linear second order boundary value problems. We mainly focus on sign changing transport functions, denoted by . There is a huge difference between the cases when is non-increasing or non-decreasing. It turns out that when is non-increasing the maximum principle holds when the amplitude of is sufficiently large, whereas it fails to be true if is non-decreasing. [Copyright &y& Elsevier]

Abstract: This paper studies the existence and the uniqueness of coexistence states in a spatially heterogeneous reaction diffusion model originated by the theory of nuclear reactors. The mathematical relevance of our analysis relies on the fact that the underlying variational systems are of non-cooperative type and, consequently, the nice comparison techniques available for cooperative systems fail to be true, making the problem of ascertaining whether or not the model admits a unique coexistence state an extremely challenging task. In this paper we are giving a series of new uniqueness results complementing those available in the literature. [Copyright &y& Elsevier]

Abstract: This paper ascertains the sharp behavior of the first zero of the radially symmetric solution of a superlinear elliptic three-dimensional equation as a function of its value at the origin, and it establishes the strict monotonicity and asymptotic finitude of the underlying associated mass. [Copyright &y& Elsevier]

We consider the Cauchy problem for a parabolic-elliptic system in $${\mathbb{R}^2}$$, which is amathematical model of chemotaxis and also amodel of self-attracting particles. In the critical mass case, we determine the basin of attraction of the steady-states for the Cauchy problem through a Lyapunov functional. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies a weighted quasilinear perturbation, through the mean curvature flow operator, of the classical linear heat equation. The mean curvature has the effect of maintaining bounded all classical positive steady states of the model, though their derivatives must be somewhere unbounded. The dynamics of the positive solutions of the model is fully described. [Copyright &y& Elsevier]

Abstract: This paper studies a quasilinear perturbation, through the mean curvature flow operator, of the classical linear heat equation. The mean curvature has the effect of maintaining bounded all classical positive steady-states of the model. [Copyright &y& Elsevier]

Abstract: This paper characterizes the existence of coexistence states in a reaction–diffusion model arising in the theory of nuclear reactors. From a mathematical point of view, the importance of this model relies upon the fact that the associated variational systems are of non-cooperative type and, consequently, the comparison techniques available for cooperative systems fail to work out. Although in higher spatial dimensions the dynamics of the model might be rather involved, by the absence of limitations for the number of steady states, we can prove the uniqueness of the steady state in the one-dimensional prototype model. Our results complement and eventually sharpen the findings of Arioli [G. Arioli, Long term dynamics of a reaction–diffusion system, J. Differential Equations 235 (2007) 298–307]. [Copyright &y& Elsevier]

This paper infers from a generalized Picone identity the uniqueness of the stable positive solution for a class of semilinear equations of superlinear indefinite type, as well as the uniqueness and global attractivity of the coexistence state in two generalized diffusive prototypes of the symbiotic and competing species models of Lotka–Volterra. The optimality of these uniqueness theorems reveals the tremendous strength of the Picone identity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we analyze the interplay between the most classical laws of Population Dynamics, Malthus versus Verhulst, in a class of sublinear cooperative parabolic systems, where, much like for one single equation, the metasolutions are imperative for describing the dynamics of the problem in the regimes where the model does not admit a stable non-negative steady-state. [Copyright &y& Elsevier]

Abstract: This paper shows the existence and the uniqueness of the positive solution of the singular boundary value problem where f is a continuous non-decreasing function such that , and h is a non-negative function satisfying the Keller–Osserman condition. Moreover, it also ascertains the exact blow-up rate of at in the special case when there exist and such that 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 for all . 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]. [Copyright &y& Elsevier]

Abstract: This paper analyzes the asymptotic behaviour as of the principal eigenvalue of the cooperative operator in a bounded smooth domain Ω of , , under homogeneous Dirichlet boundary conditions on ∂Ω, where , , and , , for all . Precisely, our main result establishes that if consists of two components, and , then where, for any and , stands for the principal eigenvalue of in D. Moreover, if we denote by the principal eigenfunction associated to , normalized so that , and, for instance, then the limit is well defined in , in and provides us with the principal eigenfunction of . This is a rather striking result, for as, according to it, the principal eigenfunction must approximate zero as if , in spite of the cooperative structure of the operator. [Copyright &y& Elsevier]

Abstract: In this paper, we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] to prove that any large solution of , , , must satisfywhereand is any smooth extension of the boundary normal section of a at , i.e.,Subsequently, stands for the outward unit normal at . Therefore, the theory can be extended to cover the general case when . [Copyright &y& Elsevier]

Abstract: This paper analyzes the dynamics of a superlinear indefinite parabolic system. As a byproduct, a number of new results related to population dynamics and economy are obtained. Among them, it is shown that the presence of refuge areas in competitive environments is an optimal mechanism to avoid extinction, and that incorporating local symbiosis in competitive environments increases productivity and allows avoiding extinction of the ‘weaker’ species. Undoubtedly, a paradigm of global markets and possibly of Earth biodiversity. Our analysis combines a series of well-known results for systems with some very recent pioneering findings within the context of superlinear indefinite equations. [Copyright &y& Elsevier]

Abstract: In this work we introduce a general class of spatially heterogeneous competing species models where the species are assumed to disperse in a random way through the inhabiting region in the presence of some refuge patches where they are free from the aggressions of the antagonist species. Our model shows that the competitive exclusion principle fails to be true under these circumstances, as the species can segregate within their respective refuge areas when the intensity of the aggressions from competitors severely increase. Going beyond, segregation mechanisms, as a result from competition, combined with subsequent species differentiation, as a consequence from territorial heterogeneities—after a certain number of generations—might ultimately explain the extraordinary biodiversity of the Earth''s biosphere, which seems to be confirmed by fossil registers in zoo-paleontology. Actually, the existence of Lazaro’ species strongly support the validity of the predictions made from our prototype model. [Copyright &y& Elsevier]

Abstract: In this work a general class of nonlinear abstract equations satisfying a generalized strong maximum principle is considered in order to study the behavior of the bounded components of positive solutions bifurcating from the curve of trivial states at a nonlinear eigenvalue with geometric multiplicity one. Since the unilateral theorems of Rabinowitz (J. Funct. Anal. 7 (1971) 487, Theorems 1.27 and 1.40) are not true as originally stated (cf. the very recent counterexample of Dancer, Bull. London Math. Soc. 34 (2002) 533), in order to get our main results the unilateral theorem of López-Gómez (Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics, vol. 426, CRC Press, Boca Raton, FL, 2001, Theorem 6.4.3) is required. Our analysis fills some serious gaps existing is some published papers that were provoked by a direct use of Rabinowitz''s unilateral theory. Actually, the abstract theory developed in this paper cannot be covered with the pioneering results of Rabinowitz (1971), since in Rabinowitz''s context any component of positive solutions must be unbounded, by a celebrated result attributable to Dancer (Arch. Rational Mech. Anal. 52 (1973) 181). [Copyright &y& Elsevier]

In this paper we use the topological degree to estimate the minimal number of solutions of the λ-slices of the semi-bounded and bounded components of a general class of fixed point equations depending on a real parameter λ by means of the signature of the component. The signature is defined as the set of bifurcation intervals from a given state together with all changes of the local index of the state. Calculating the minimal number of solutions of the component slices enables us to ascertain its minimal complexity in terms of graph theory, at least for a generic family of non-degenerate situations, where the general theory of this paper provides the minimal (canonical) structure of the component. [Copyright &y& Elsevier]

In this paper we ascertain the blow-up rate of the large solutions of a class of sublinear elliptic boundary value problems with a weight function in front of the nonlinearity that vanishes on the boundary of the underlying domain, Ω, at different rates according to the point of the boundary, x∞∈∂Ω. All previous results in the literature assumed the decay rate of the underlying weight function to be the same at any point of ∂Ω. This hypothesis substantially simplified the mathematical analysis of the problem, as it allowed constructing global sub and supersolutions in an open neighborhood of ∂Ω. Obtaining general results requires localizing at each particular point of the boundary, making particularly involved the mathematical analysis of the problem. [Copyright &y& Elsevier]

This paper analyzes the limiting behavior of the positive solutions of a general class of sublinear elliptic weighted mixed boundary value problems as the amplitude of the positive part of the lower order terms of the differential operator blows up to infinity. The main result establishes that the positive solutions approximate zero within the support of the positive part of the potential, whereas they stabilize to the positive solution of a certain elliptic mixed boundary value problem on its complement. Further, we use this result for deriving some general principles in competing species dynamics. Precisely, we shall show that in the presence of a refuge region two competing species must coexist if their reproduction rates are sufficiently large, independently of the strength of the competition. It should be emphasized that the abstract theory developed here allows measuring how large the reproduction rates should be for being permanent, providing us, simultaneously, with the limiting behavior of each of the species separately. Basically, when the pressure from the competitor grows the tackled species concentrates within its refuge. The results of this paper are substantial extensions of some pioneer results found by one of the authors in [16, Section 4]. The main ingredients in deriving the main results of this paper are the continuous dependence of the principal eigenvalue with respect to a general class of perturbations of the domain around its Dirichlet boundary – very recent result coming from [6] – and the continuous dependence of the positive solutions of the sublinear problem – coming from [7]. [Copyright &y& Elsevier]

In this paper we use the linear theory developed in Cano-Casanova and Lo´pez-Go´mez (J. Differential Equations 178 (2002) 123) to show the continuous dependence of the positive solutions of a general class of sublinear elliptic boundary value problems of mixed type with respect to the boundary operator. Besides working with mixed non-classical boundary conditions, this paper considers problems where bifurcation from infinity occurs. Treating this kind of problems is far from easy, since a complete knowledge of the set of values of the parameters where they exhibit a positive solution is imperative. Our results are far from immediate, as a consequence of the vanishing weight functions involved in our general setting. Now a days, the analysis of these general problems, where the coefficients are allowed to vary and eventually vanishing or changing sign, is focusing a great deal of attention. [Copyright &y& Elsevier]

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 Equations5, 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 Equations138, 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. Lo´pez-Go´mez (1998, J. Differential Equations146, 336–374). [Copyright &y& Elsevier]

This paper tries to establish a link between topological and algebraic methods in nonlinear analysis showing how the topological degree for Fredholm operators of index zero of Fitzpatrick et al. (1994) can be determined from the generalized algebraic multiplicity of Esquinas and López-Gómez (1988, 2001), in the same vein as the Leray–Schauder degree can be calculated from the Schauder formula through the classical algebraic multiplicity. [ABSTRACT FROM AUTHOR]