On sign-changing and multiple solutions of the p-Laplacian

In this paper, we construct the pseudo-gradient vector field in , by which we obtain the positive and negative cones of are both invariant sets of the descent flow of the corresponding functional. Then we use differential equations theory in Banach spaces and dynamics theory to study p-Laplacian boundary value problems with “jumping” nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems of p-Laplacian.     

On finding sign-changing solutions

Some parameter-depending linking theorems are established, which allow to produce a bounded and sign-changing Palais-Smale sequence. For even functionals, a parameter-depending fountain theorem is obtained which provides infinitely many bounded and sign-changing Palais-Smale sequences. A variant mountain pass theorem is built in cones which yields bounded, positive and negative Palais-Smale sequences. The usual Palais-Smale type compactness condition and its variants are completely not necessary for these theories. More exact locations of the critical sequences can be determined. The abstract results are applied to the Schrödinger equation with (or without) critical Sobolev exponents:

     

On sign-changing solutions for nonlinear operator equations

In this paper, the existence of sign-changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory. The main theorems are some new three-solution theorems which are different from the famous Amann's and Leggett-Williams' three-solution theorems as well as the results in [F. Li, G. Han, Generalization for Amann's and Leggett-Williams' three-solution theorems and applications, J. Math. Anal. Appl. 298 (2004) 638-654]. These three solutions are all nonzero. One of them is positive, another is negative, and the third one is a sign-changing solution. Furthermore, the theoretical results are successfully applied to both integral and differential equations.     

The existence of multiple positive solutions for singular functional differential equations with sign-changing nonlinearity

In this paper, we study the existence of multiple positive solutions for boundary value problems based on second-order functional differential equations with the form

     

Constant-sign and sign-changing solutions for nonlinear eigenvalue problems

We prove the existence of multiple constant-sign and sign-changing solutions for a nonlinear elliptic eigenvalue problem under Dirichlet boundary condition involving the p$p$-Laplacian. More precisely, we establish the existence of a positive solution, of a negative solution, and of a nontrivial sign-changing solution when the eigenvalue parameter λ$\lambda$ is greater than the second eigenvalue _λ2${\lambda }_2$ of the negative p$p$-Laplacian, extending results by Ambrosetti–Lupo, Ambrosetti–Mancini, and Struwe. Our approach relies on a combined use of variational and topological tools (such as, e.g., critical points, Mountain-Pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the p$p$-Laplacian) and comparison arguments for nonlinear differential inequalities. In particular, the existence of extremal nontrivial constant-sign solutions plays an important role in the proof of sign-changing solutions.     

Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian

Existence of multiple and sign-changing solutions for a problem involving p-Laplacian and jumping nonlinearities are considered via the construction of descent flow in . Sign-changing and multiple solutions are obtained under additional assumption on the nonlinearity. The uniqueness of positive (negative) solution theorem is included too.     

Existence of multiple and sign-changing solutions for a second-order nonlinear functional difference equation with periodic coefficients

In the present paper, we apply the method of invariant sets of descending flow to establish a series of criteria to ensure that a second-order nonlinear functional difference equation with periodic boundary conditions possesses at least one trivial solution and three nontrivial solutions. These nontrivial solutions consist of sign-changing solutions, positive solutions and negative solutions. Moreover, as an application of our theoretical results, an example is elaborated. Our results generalize and improve some existing ones.     

Positive and sign-changing solutions for fourth-order BVPs with parameters

This paper is concerned with the existence and multiplicity of positive and sign-changing solutions of the fourth-order boundary value problem u ⁽⁴⁾(t)=λ f(t,u(t),u ^′′(t)), 0<t<1,?u(0)?=?u(1)=u ^′′(0)=u ^′′(1)?=0, where f:[0,1]×?→? is continuous, λ∈? is a parameter. By using the fixed-point index theory of differential operators, it is proved that the above boundary value problem has positive, negative and sign-changing solutions for λ being different intervals. As an example, the boundary value problem u ⁽⁴⁾(t)+?η u ^′′(t)??ζu(t)=?λ f(t,u(t)), ?0<t<1,?u(0)=?u(1)=?u ^′′(0)=?u ^′′(1)=0 is also considered and some obtained results are the complement of the known results.     

Exact solutions for nonlinear Burgers' equation by homotopy perturbation method

The aim of this article is to construct a new efficient recurrent relation to solve nonlinear Burgers' equation. The homotopy perturbation method is used to solve this equation. Because Burgers' equation arises in many applications, it is worth trying new solution methods. Comparison of the results with those of Adomian's decomposition method leads to significant consequences. Four standard problems are used to illustrate the method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Nonexistence of sign-changing solutions for some elliptic and parabolic inequalities

We establish nonexistence of nontrivial solutions (including sign-changing ones) for some partial differential inequalities of elliptic and parabolic type containing nonlinear terms that depend on the positive and negative part of the sought function in different ways. Systems of elliptic inequalities with similar structure are also considered. The proofs are based on the test function method.     

Multiple positive solutions for semi-linear elliptic systems with sign-changing weight

In this paper, we study the multiplicity results of positive solutions for a semi-linear elliptic system involving both concave–convex and critical growth terms. With the help of the Nehari manifold and the Lusternik–Schnirelmann category, we investigate how the coefficient h(x)$h\left(x\right)$ of the critical nonlinearity affects the number of positive solutions of that problem and get a relationship between the number of positive solutions and the topology of the global maximum set of h$h$.