On a Class of Quasilinear Elliptic Equations

We consider a class of quasilinear elliptic boundary problems, including the following Modified Nonlinear Schrödinger Equation as a special case: $$\begin{cases} ∆u+ \frac{1}{2} u∆(u^2)−V(x)u+|u|^{q−2}u=0 \ \ \ in \ Ω, \\u=0 \ \ \ \ \ \ \ ~ ~ ~ on \ ∂Ω, \end{cases}$$ where $Ω$ is the entire space $\mathbb{R}^N$ or $Ω ⊂ \mathbb{R}^N$ is a bounded domain with smooth boundary, $q∈(2,22^∗]$ with $2^∗=2N/(N−2)$ being the critical Sobolev exponent and $22^∗= 4N/(N−2).$ We review the general methods developed in the last twenty years or so for the studies of existence, multiplicity, nodal property of the solutions within this range of nonlinearity up to the new critical exponent $4N/(N−2),$ which is a unique feature for this class of problems. We also discuss some related and more general problems.     

Equivalence Relation between Initial Values and Solutions for Evolution p-Laplacian Equation in Unbounded Space

In this paper,an equivalence relation between the ω-limit set of initial values and the ω-limit set of solutions is established for the Cauchy problem of evolution p-Laplacian equation in the unbounded space Yσ(RN).To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions,we establish the propagation estimate and the growth estimate for the solutions.It will be demonstrated that the equivalence relation can be used to study the asymptotic behavior of solutions.     

Study on a kind of $p$-Laplacian neutral differential equation with multiple variable coefficients

In this paper, we first discuss some properties of the neutral operator with multiple variable coefficients $(Ax)(t):=x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)$. Afterwards, by using an extension of Mawhin''s continuation theorem, a kind of second order $p$-Laplacian neutral differential equation with multiple variable coefficients as follows $$\left(\phi_p\left(x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)\right)''\right)''=\tilde{f}(t,x(t),x''(t))$$ is studied. Finally, we consider the existence of periodic solutions for two kinds of second-order $p$-Laplacian neutral Rayleigh equations with singularity and without singularity. Some new results on the existence of periodic solutions are obtained. It is worth noting that $c_i$ ($i=1,\cdots,n$) are no longer constants which are different from the corresponding ones of past work.     

Removable sets for continuous solutions of quasilinear elliptic equations

We show that sets of Hausdorff measure zero are removable for -Hölder continuous solutions to quasilinear elliptic equations similar to the -Laplacian. The result is optimal. We also treat larger sets in terms of a growth condition. In particular, our results apply to quasiregular mappings.

     

On the eigenvalues of the

We study the nonlinear eigenvalue problem

where , is a bounded smooth domain in . We prove that the first and the second variational eigenvalues of (1) are continuous functions of . Moreover, we obtain the asymptotic behavior of the first eigenvalue as and .

     

Bifurcation problems for the

In this paper we consider the bifurcation problem

in with . We show that a continuum of positive solutions bifurcates out from the principal eigenvalue of the problem

Here both functions and may change sign.

     

Degenerate Nonlinear Elliptic Equations Lacking in Compactness

In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form-div(a(x)▽_p u(x)) = g(λ, x, |u|~(p-2)u) in R~N, where ▽_pu =|▽u|~(p-2)▽u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.     

一类具$p$-Laplace非线性三点边值问题三个正解的存在性

This paper deals with the existence of three positive solutions for a class of nonlinear singular three-point boundary value problem with p-Laplacian. By means of a fixed point theorem duo to Leggett and Williams, sufficient condition for the existence of at least three positive solutions to the nonlinear singular three-point boundary value problem is established     

带$p$-Laplacian的四阶四点奇异边值问题的正解

In this paper, we consider the existence of positive solutions for the singular fourthorder four point boundary value problem with p-Laplacian operator. By using the fixed point theorem of cone expansion and compression, the existence of multiple positive solutions is obtained.     

半拟线性p-Laplacian问题结点解的存在性

In this paper, we study the existence of nodal solutions for the following problem:-(φ_p(x′))′= α(t)φ_p(x~+) + β(t)φ_p(x~-) + ra(t)f(x), 0 t 1,x(0) = x(1) = 0,where φ_p(s) = |s|~(p-2)s, a ∈ C([0, 1],(0, ∞)), x~+= max{x, 0}, x~-=- min{x, 0}, α(t), β(t) ∈C[0, 1]; f ∈ C(R, R), sf(s) 0 for s ≠ 0, and f_0, f_∞∈(0, ∞), where f_0 = lim_|s|→0f(s)/φ_p(s), f_∞ = lim|s|→+∞f(s)/φ_p(s).We use bifurcation techniques and the approximation of connected components to prove our main results.