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1.
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. 相似文献
2.
Equivalence Relation between Initial Values and Solutions for Evolution p-Laplacian Equation in Unbounded Space 下载免费PDF全文
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. 相似文献
3.
Tero Kilpelä inen Xiao Zhong 《Proceedings of the American Mathematical Society》2002,130(6):1681-1688
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.
4.
Study on a kind of $p$-Laplacian neutral differential equation with multiple variable coefficients 下载免费PDF全文
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. 相似文献
5.
Yin Xi Huang 《Proceedings of the American Mathematical Society》1997,125(11):3347-3354
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 .
6.
Pavel Drá bek Yin Xi Huang 《Transactions of the American Mathematical Society》1997,349(1):171-188
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.
7.
In this paper, the authors obtain the gradient estimates for positive solutions to the weighted p-Laplacian Lichnerowicz equation △p,fu + cuσ= 0 on noncompact smooth metric measure space, where c is a nonnegative constant, and p, σ(1 < p≤2, σ≤p-1) are real constants. Moreover, by the gradient estimate, they can get the corresponding Liouville theorem and Harnack inequality. 相似文献
8.
Paul Binding Lyonell Boulton Jan Cepicka Pavel Drá bek Petr Girg 《Proceedings of the American Mathematical Society》2006,134(12):3487-3494
For , the eigenfunctions of the non-linear eigenvalue problem for the -Laplacian on the interval are shown to form a Riesz basis of and a Schauder basis of whenever .
9.
Positive solutions for a nonlinear discrete fractional boundary value problem with a $p$-Laplacian operator 下载免费PDF全文
Wei Cheng Jiafa Xu Donal O Regan Donal O''Regan Yujun Cui 《Journal of Applied Analysis & Computation》2019,9(5):1959-1972
In this paper using the monotone iterative technique we establish the existence and uniqueness of positive solutions for a nonlinear discrete fractional boundary value problem with a $p$-Laplacian operator. Also we discuss an iterative sequence which yields the approximate solution for this problem. 相似文献
10.
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. 相似文献