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.     

Stationary layered solutions in {\mathbb{R}}^2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations where is a multiple-well potential and is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as corresponding to minima of W. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation the equation (0.1) has solutions which depends on both the variables x andy. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x. Received April 16, 1999 / Accepted October 1, 1999 / Published online June 28, 2000     

Multiple positive and sign-changing solutions for a singular Schrödinger equation with critical growth

Variational methods are used to prove the existence of multiple positive and sign-changing solutions for a Schrödinger equation with singular potential having prescribed finitely many singular points. Some exact local behavior for positive solutions obtained here are also given. The interesting aspects are two. One is that one singular point of the potential V(x)$V(x)$ and one positive solution can produce one sign-changing solution of the problem. The other is that each sign-changing solution changes its sign exactly once.     

A priori estimates for the scalar curvature equation on <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

We obtain a priori estimates for solutions to the prescribed scalar curvature equation on S ³. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse function.     

On the existence of pair positive–negative solutions for resonance problems

This paper deals with a class of semilinear elliptic Dirichlet boundary value problems at resonance. We introduce a sufficient Landesman–Lazer condition for the existence of pair positive–negative solutions. Furthermore, developing the fibering method in the framework of the Leray–Schauder degree theory we can prove the existence of branches for positive and negative solutions.     

Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian

We consider a nonlinear elliptic equation driven by the p-Laplacian with Dirichlet boundary conditions. Using variational techniques combined with the method of upper-lower solutions and suitable truncation arguments, we establish the existence of at least five nontrivial solutions. Two positive, two negative and a nodal (sign-changing) solution. Our framework of analysis incorporates both coercive and p-superlinear problems. Also the result on multiple constant sign solutions incorporates the case of concave-convex nonlinearities.     

On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux

Under some non-degeneracy condition we show that sequences of entropy solutions of a semi-linear elliptic equation are strongly pre-compact in the general case of a Carathéodory flux vector. The proofs are based on localization principles for H-measures corresponding to sequences of measure-valued functions.     

Structure of the solution set for a class of semilinear elliptic equations with asymptotic linear nonlinearity

We consider a semilinear elliptic equation with asymptotic linear nonlinearity applying bifurcation theory and spectral analysis. We obtain the exact multiplicity of the positive solutions and a very precise structure of the solution set, which improves the previous knowledge of the problem.     

Positive solutions of semilinear biharmonic equations with critical Sobolev exponents

In this paper we study the critical growth biharmonic problem with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the biharmonic problem.     

Solutions with sign information for nonlinear nonhomogeneous problems

We consider a parametric elliptic equation driven by a nonlinear nonhomogeneous differential operator and with a Robin boundary condition. In the first part we prove the existence of positive solutions and state a bifurcation‐type result describing how the set of positive solutions changes as the parameter varies. In the second part we show that problem admits nodal (sign‐changing) solutions provided the parameter is sufficiently large.     

Symmetry breaking in a bounded symmetry domain

In this article, we prove that there are three positive solutions of a semilinear elliptic equation in a bounded symmetric domain _t for large t>0 in which one is axially symmetric and the other two are nonaxially symmetric. Main tools are the Palais-Smale theory.     

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

In this article, we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation technique. Our results complete and improve a work published recently by Zhang and Zhou(existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Applied Mathematics Letters,Volume 50, December 2015, Pages 48–55).     

Existence of three nontrivial solutions for nonlinear Neumann hemivariational inequalities

In this paper we consider a nonlinear Neumann problem driven by the p$p$-Laplacian with a nonsmooth potential (hemivariational inequality). Using minimax methods based on the nonsmooth critical point theory together with suitable truncation techniques, we show that the problem has at least three nontrivial smooth solutions. Two of these solutions have constant sign (one is positive, the other negative).     

Multiple Positive Solutions for Eigenvalue Problems of Hemivariational Inequalities

We study a nonlinear eigenvalue problem with a nonsmooth potential. The subgradients of the potential are only positive near the origin (from above) and near +∞. Also the subdifferential is not necessarily monotone (i.e. the potential is not convex). Using variational techniques and the method of upper and lower solutions, we establish the existence of at least two strictly positive smooth solutions for all the parameters in an interval. Our approach uses the nonsmooth critical point theory for locally Lipschitz functions. A byproduct of our analysis is a generalization of a result of Brezis-Nirenberg (CRAS, 317 (1993)) on H¹₀ versus C¹₀ minimizers of a C¹-functional.     

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.     

Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations

In this paper we are concerned with multi-lump bound states of the nonlinear Schr?dinger equation for sufficiently small , where for and for . V is bounded on . For any finite collection of nondegenerate critical points of V, we show the uniqueness of solutions of the form for , where u is positive on and is a small perturbation of a sum of one-lump solutions concentrated near , respectively for sufficiently small . Received: 30 October 2001; in final form: 10 June 2002 /Published online: 2 December 2002 RID="*" ID="*" Research supported by Alexander von Humboldt Foundation in Germany and NSFC in China     

Existence and regularity for a nonlinear boundary flow problem of population genetics

In this paper we consider a heat equation with nonlinear boundary condition occurring in population genetics, the selection–migration problem for alleles in a region, considering flow of genes throughout the boundary. Such a problem determines a gradient flow in a convenient phase space and then the dynamics for large times depends heavily on the knowledge of the equilibrium solutions. We address the questions of the existence of a nontrivial equilibrium solution and its regularity.     

Singularly perturbed elliptic systems

We study the existence and concentration behavior of positive solutions for a class of Hamiltonian systems (two coupled nonlinear stationary Schrödinger equations). Combining the Legendre–Fenchel transformation with mountain pass theorem, we prove the existence of a family of positive solutions concentrating at a point in the limit, where related functionals realize their minimum energy. In some cases, the location of the concentration point is given explicitly in terms of the potential functions of the stationary Schrödinger equations.     

Small perturbations of an indefinite elliptic equation

We investigate a perturbed semilinear indefinite elliptic equation and show that the results known for the unperturbed equation still hold if the perturbation is sufficiently small. To this end, we use a continuity argument that allows us to establish the existence of two positive solutions even in the case where the strong maximum principle does not apply.     

Multiple solutions for a semilinear equation involving singular potential and critical exponent

Variational methods are used to prove the existence of positive and sign-changing solutions for a semilinear equation involving singular potential and critical exponent in any bounded domain.*supported in part by Tian Yuan Foundation of NNSF (A0324612)**Supported by 973 Chinese NSF and Foundation of Chinese Academy of Sciences.***Supported in part by NNSF of China.Received: September 23, 2002; revised: November 30, 2003