Dirichlet problems with double resonance and an indefinite potential

We consider semilinear Dirichlet problems with an unbounded and indefinite potential and with a Carathéodory reaction. We assume that asymptotically at infinity the problem exhibits double resonance. Using variational methods, together with Morse theory and flow invariance arguments, we prove multiplicity theorems producing three, five, six or seven nontrivial smooth solutions. In most multiplicity theorems, we provide precise sign information for all the solutions established.     

Multibump nodal solutions for an indefinite superlinear elliptic problem

We define some Nehari-type constraints using an orthogonal decomposition of the Sobolev space and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem.     

On the existence of postive solutions for some indefinite superlinear elliptic problems

In this work we show the existence and stability of positive solutions for a general calss of semilinear elliptic boundary value problems of superlinear type with indedefinite weight functions. Optimal necessary and sufficient conditions are found.     

On elliptic problems with indefinite superlinear boundary conditions

In this paper we prove the existence of positive solutions to some nondivergent elliptic equations with indefinite nonlinear boundary conditions. The proof is based on a new Liouville-type theorem about the nonnegative solutions to some canonical indefinite elliptic equations, which is also proved in this paper by the method of moving planes.     

Positive solutions of superlinear semipositone singular Dirichlet boundary value problems

In this paper, we study a class of superlinear semipositone singular second order Dirichlet boundary value problem. A sufficient condition for the existence of positive solution is obtained under the more simple assumptions.     

Positive solutions for Neumann problems with indefinite and unbounded potential

We consider semilinear Neumann equations with an indefinite and unbounded potential. We establish the existence and uniqueness of positive solutions. We show that our setting incorporates as special cases several parametric equations of interest (such as the equidiffusive logistic equation).     

Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

Multiple positive solutions of superlinear elliptic problems with sign-changing weight

We study the existence of multiple positive solutions for a superlinear elliptic PDE with a sign-changing weight. Our approach is variational and relies on classical critical point theory on smooth manifolds. A special care is paid to the localization of minimax critical points.     

Nontrivial solutions for Neumann problems with an indefinite linear part

In this paper, we consider a semilinear Neumann problem with an indefinite linear part and a Carathéodory nonlinearity which is superlinear near infinity and near zero, but does not satisfy the Ambrosetti-Rabinowitz condition. Using an abstract existence theorem for C¹-functions having a local linking at the origin, we establish the existence of at least one nontrivial smooth solution.     

Positive solutions for nonlinear Robin problems with indefinite potential and competing nonlinearities

Positivity - We consider a nonlinear Robin problem associated to the p-Laplacian plus an indefinite potential. In the reaction we have the competing effects of two nonlinear terms. One is...     

A priori bounds and complete blow-up of positive solutions of indefinite superlinear parabolic problems

We study a priori estimates of positive solutions of the equation t_∂u−Δu=λu+a(x)up, x∈Ω, t>0, satisfying the homogeneous Dirichlet boundary conditions. Here Ω is a bounded domain in Rn, λ∈R, p>1 is subcritical, changes sign and a,p satisfy some additional technical hypotheses. Assume that the solution u blows up in a finite time T and the set is connected. Using our a priori bounds, we show that u blows up completely in Ω⁺ at t=T and the blow-up time T depends continuously on the initial data.     

Multiple positive solutions for superlinear second order singular boundary value problems with derivative dependence

The existence of at least two positive solutions is presented for the singular second-order boundary value problem
{1/p（t）（ p（t）x′（t））′＋Φ（t）f（t,x（t）,p（t）x′（t））=0,0〈t〈1,
limt→0 p（t）x′（t）=0,x（1）=0
by using the fixed point index, where f may be singular at x = 0 and px ′= 0.     

On the multiplicity of radial solutions to superlinear Dirichlet problems in bounded domains

In this paper we are concerned with the existence and multiplicity of nodal solutions to the Dirichlet problem associated to the elliptic equation Δu+q(|x|)g(u)=0 in a ball or in an annulus in .The nonlinearity g has a superlinear and subcritical growth at infinity, while the weight function q is nonnegative in [0,1] and strictly positive in some interval [r₁,r₂]⊂[0,1].By means of a shooting approach, together with a phase-plane analysis, we are able to prove the existence of infinitely many solutions with prescribed nodal properties.     

Multiple solutions for indefinite functionals and applications to asymptotically linear problems

In this paper, by means of Morse theory of isolated critical points (orbits) we study further the critical points theory of asymptotically quadratic functionals and give some results concerning the existence of multiple critical points (orbits) which generalize a series of previous results due to Amann, Conley, Zehnder and K.C. Chang. As applications, the existence of multiple periodic solutions for asymptotically linear Hamiltonian systems is investigated. And our results generalize some recent ones due to Coti-Zelati, J.Q.Liu, S.Li, etc.This research was supported in part by the National Postdoctoral Science Fund.     

Multiple equilibria,periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation in , where $\Omega$ is a bounded domain in and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.     

Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces

The existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper. Our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of Leray-Schauder type and on a fixed point theorem in cones.     

On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order

This paper is concerned with the multiplicity of radially symmetric solutions to the Dirichlet problem

on the unit ball with boundary condition on . Here is a positive function and is a function that is superlinear (but of subcritical growth) for large positive , while for large negative we have that , where is the smallest positive eigenvalue for in with on . It is shown that, given any integer , the value may be chosen so large that there are solutions with or less interior nodes. Existence of positive solutions is excluded for large enough values of .

     

Liouville theorems,a priori estimates,and blow-up rates for solutions of indefinite superlinear parabolic problems

In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.     

A note on a superlinear indefinite Neumann problem with multiple positive solutions

We prove the existence of three positive solutions for the Neumann problem associated to u^″+a(t)uγ+1=0, assuming that a(t) has two positive humps and is large enough. Actually, the result holds true for a more general class of superlinear nonlinearities.