Bifurcation from trivial solution for elliptic systems with nonlinear boundary conditions

In this paper, we investigate semilinear elliptic systems having a parameter with nonlinear Neumann boundary conditions over a smooth bounded domain. The objective of our study is to analyse bifurcation component of positive solutions from trivial solution and their stability. The results are obtained via classical bifurcation theorem from a simple eigenvalue, by studying the eigenvalue problem of elliptic systems.     

Local and global bifurcation results for a semilinear boundary value problem

We investigate the local and global nature of the bifurcation diagrams which can occur for a semilinear elliptic boundary value problem with Neumann boundary conditions involving sign-changing coefficients. It is shown that closed loops of positive and negative solutions occur naturally for such problems and properties of these loops are investigated.     

On eigenvalue problems with Robin type boundary conditions having indefinite coefficients

A Robin type boundary condition with a sign-changing coefficient is treated. First, the associated linear elliptic eigenvalue problem is studied, where the existence of a principal eigenvalue is discussed by the use of a variational approach. Second, the associated semilinear elliptic boundary value problem of logistic type is studied and the one parameter-dependent structure of positive solutions is investigated, where results obtained are due to the construction of suitable super- and subsolutions by using the principal positive eigenfunctions of the linear eigenvalue problem.     

Bifurcation and stability results for an equation arising in population genetics

Summary We consider a semilinear elliptic boundary value problem, which arises in population genetics. Although there is no obvious corresponding linearized problem, we establish, by using Implicit Function Theorem methods, necessary and sufficient conditions forbifurcation to occur from a branch of trivial solutions. The stability of the bifurcating solutions is also investigated.     

Bifurcation approach to a logistic elliptic equation with a homogeneous incoming flux boundary condition

In this paper, we consider a semilinear elliptic boundary value problem in a smooth bounded domain, having the so-called logistic nonlinearity that originates from population dynamics, with a nonlinear boundary condition. Although the logistic nonlinearity has an absorption effect in the problem, the nonlinear boundary condition is induced by the homogeneous incoming flux on the boundary. The objective of our study is to analyze the existence of a bifurcation component of positive solutions from trivial solutions and its asymptotic behavior and stability. We perform this analysis using the method developed by Lyapunov and Schmidt, based on a scaling argument.     

Nonlinear elliptic problems with superlinear reaction and parametric concave boundary condition

We study parametric nonlinear elliptic boundary value problems driven by the p-Laplacian with convex and concave terms. The convex term appears in the reaction and the concave in the boundary condition (source). We study the existence and nonexistence of positive solutions as the parameter λ > 0 varies. For the semilinear problem (p = 2), we prove a bifurcation type result. Finally, we show the existence of nodal (sign changing) solutions.     

Symmetry-breaking at positive solutions

We consider symmetry-breaking bifurcations at positive solutions of semilinear elliptic boundary value problems in a ball, with Dirichlet boundary conditions. We prove that symmetry-breaking can only occur by bifurcation of axisymmetric solutions. We also obtain sufficient conditions for such symmetry-breaking.Dedicated to Professor H. Knobloch at the occasion of his 60th birthday     

The effects of indefinite nonlinear boundary conditions on the structure of the positive solutions set of a logistic equation

We investigate a semilinear elliptic equation with a logistic nonlinearity and an indefinite nonlinear boundary condition, both depending on a parameter λ. Overall, we analyze the effect of the indefinite nonlinear boundary condition on the structure of the positive solutions set. Based on variational and bifurcation techniques, our main results establish the existence of three nontrivial non-negative solutions for some values of λ, as well as their asymptotic behavior. These results suggest that the positive solutions set contains an S-shaped component in some case, as well as a combination of a C-shaped and an S-shaped components in another case.     

Symmetry and monotonicity of positive solutions of elliptic equations with mixed boundary conditions in a super-spherical cone

In this paper, we study symmetry and monotonicity properties for positive solutions of semilinear elliptic equations with mixed boundary conditions. We establish a version of the maximum principle for mixed boundary conditions in a narrow domain, and prove the positive solution in super-spherical cone is axially symmetric and monotone in some direction by the methods of moving planes.     

Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight

We study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear Neumann boundary condition having a parameter and an indefinite weight. The main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary, introduced by Fisher and Fleming, and Henry?s approach to the problem.Local and global structures of the set of equilibria are given. While the stability of constant equilibria is analyzed, the exponential stability of the unique bifurcating nonconstant equilibrium solution is established. Diagrams exhibiting the bifurcation and stability structures are also furnished. Moreover the asymptotic behavior of such solutions on the boundary of the domain, as the positive parameter goes to infinity, is also provided.The results are obtained via classical tools like the Implicit Function Theorem, bifurcation from a simple eigenvalue theorem and the exchange of stability principle, in a combination with variational and dynamical arguments.     

Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions

Three important inequalities (the Poincaré, Hardy and generalized Poincaré inequalities) on the mixed boundary conditions are firstly established by some analytical techniques. Then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed Dirichlet-Neumann boundary conditions involving Hardy terms and Hardy-Sobolev critical exponents by using the variational methods.     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

Multiple and sign changing solutions of an elliptic eigenvalue problem with constraint

We prove the existence of sign changing solutions of a semilinear elliptic eigenvalue problem with constraint by using variational methods. Among those three solutions we obtained, one is positive, one negative and one sign changing. We also prove the existence of multiple sign changing solutions under some additional condition.     

Mathematics

We prove the existence of sign changing solutions of a semilinear elliptic eigenvalue problem with constraint by using variational methods. Among those three solutions we obtained, one is positive, one negative and one sign changing. We also prove the existence of multiple sign changing solutions under some additional condition.     

Sequential and continuum bifurcations in degenerate elliptic equations

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.

     

Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth

We consider a classical semilinear elliptic equation with Neumann boundary conditions on an annulus in R ^N . The nonlinear term is the product of a radially symmetric coefficient with a pure power. We prove that if the power is sufficiently large, the problem admits at least three distinct positive and radial solutions. In case the coefficient is constant, we show that none of the three solutions is constant. The methods are variational and are based on the study of a suitable limit problem.     

Global solution curves for a class of elliptic systems

We study curves of positive solutions for a system of elliptic equations of Hamiltonian type on a unit ball. We give conditions for all positive solutions to lie on global solution curves, allowing us to use the analysis similar to the case of one equation, as developed in P. Korman, Y. Li and T. Ouyang [An exact multiplicity result for a class of semilinear equations, Commun. PDE 22 (1997), pp. 661–684.] (see also T. Ouyang and J. Shi [Exact multiplicity of positive solutions for a class of semilinear problems, II, J. Diff. Eqns. 158(1) (1999), pp. 94–151].). As an application, we obtain some non-degeneracy and uniqueness results. For the one-dimensional case we also prove the positivity for the linearized problem, resulting in more detailed results.     

Exact multiplicity and global structure of solutions for a class of semilinear elliptic equations

In this paper, we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation. We also obtain a precise global bifurcation diagram of the solution set. As a result, an open problem presented by C.-H. Hsu and Y.-W. Shih [C.-H. Hsu, Y.-W. Shih, Solutions of semilinear elliptic equations with asymptotic linear nonlinearity, Nonlinear Anal. 50 (2002) 275-283] is completely solved. Our argument is mainly based on bifurcation theory and continuation method.     

Gelfand type elliptic problems under Steklov boundary conditions

For a Gelfand type semilinear elliptic equation we extend some known results for the Dirichlet problem to the Steklov problem. This extension requires some new tools, such as non-optimal Hardy inequalities, and discovers some new phenomena, in particular a different behavior of the branch of solutions and three kinds of blow-up for large solutions in critical growth equations. We also show that small values of the boundary parameter play against strong growth of the nonlinear source.