Neumann Problem of Quasilinear Elliptic Equations with Limit Nonlinearity in Boundary Condition

This paper is concerned with the existence of solutions to the equation D_j(a^{ij}(x,u)D_i,u)-\frac{1}{2}D_sa^{ij}(x,u)D_iuD_ju + λ u = 0 on a bounded domain under the Neumann boundary condition a{ij}(x,u)D_iuϒ_j = lul^{\frac{2}{n-2}}u.     

Neumann problem for generalized n-Poisson equation

Using a hierarchy of integral operators having higher-order Neumann functions and their derivatives as kernels, the Neumann problem for a 2nth order linear partial complex differential equation is discussed. The solvability of the problem is obtained.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp

In this work we analyze the existence and regularity of the solution of a nonhomogeneous Neumann problem for the Poisson equation in a plane domain Ω with an external cusp. In order to prove that there exists a unique solution in H¹(Ω) using the Lax–Milgram theorem we need to apply a trace theorem. Since Ω is not a Lipschitz domain, the standard trace theorem for H¹(Ω) does not apply, in fact the restriction of H¹(Ω) functions is not necessarily in L²(∂Ω). So, we introduce a trace theorem by using weighted Sobolev norms in Ω. Under appropriate assumptions we prove that the solution of our problem is in H²(Ω) and we obtain an a priori estimate for the second derivatives of the solution.     

Uniqueness of the least-energy solution for a semilinear Neumann problem

We prove that the least-energy solution of the problem

where is a ball, and if , if , is unique (up to rotation) if is small enough.

     

Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity

In this paper, we study the existence of ground state solutions of nonlinear elliptic equation with logarithmic nonlinearity by the Linking theorem and logarithmic Sobolev inequality. Our results are quite different from those in the case of polynomial nonlinearity.     

Multiple existence of solutions for a semilinear elliptic problem with Neumann boundary condition

The multiplicity of solutions for semilinear elliptic equations with exponential growth nonlinearities is treated. The approach to the problem is a variational method.     

On a generalization of the Neumann problem for the Laplace equation outside cuts in a plane

A boundary value problem for harmonic functions outside cuts in a plane is considered. The jump of the normal derivative is specified on the cuts as well as a linear combination of the normal derivative on one side of the cut and the jump of the unknown function. The problem is studied with three different conditions at infinity, which lead to different results on existence and number of solutions. The integral representation for a solution is obtained in the form of potentials density in which satisfies the uniquely solvable Fredholm integral equation of the 2nd kind. Copyright © 2004 John Wiley & Sons, Ltd.     

A sub-supersolution approach for Neumann boundary value problems with gradient dependence

Existence and location of solutions to a Neumann problem driven by an nonhomogeneous differential operator and with gradient dependence are established developing a non-variational approach based on an adequate method of sub-supersolution. The abstract theorem is applied to prove the existence of finitely many positive solutions or even infinitely many positive solutions for a class of Neumann problems.     

具临界指数和临界非线性边界条件的拟线性椭圆型方程Neumann问题

本文给出ＲＮ（Ｎ３）中有界光滑区域Ω上的拟线性椭圆型方程：－∑Ｎｉ＝１ｘｉ·｜Ｄｕ｜ｐ－２ｕｘｉ＝λ｜ｕ｜ｐ－２ｕ＋ａ（ｘ）｜ｕ｜ｐ－２ｕ＋ｆ（ｘ，ｕ），ｘ∈Ω（λ＞０，ｐ＝Ｎｐ／（Ｎ－ｐ），２ｐ＜Ｎ）在边界条件：－｜Ｄｕ｜ｐ－２Ｄνｕ｜Ω＝ψ（ｘ）｜ｕ｜ｑ－２ｕ（ｑ＝（Ｎ－１）ｐ／（Ｎ－ｐ））下的多解性结果．     

A quasilinear Neumann problem with discontinuous nonlinearity

In this paper we study a quasilinear boundary value problem of Neumann type with discontinuous terms. In particular we consider a problem in which the p–Laplacian is involved. In order to prove the existence of solutions we replace this problem with a multivalued approximation of it and, using a variational approach for locally Lipschitz functionals, we prove two existence results for it.     

On a singular parabolic p-biharmonic equation with logarithmic nonlinearity

This paper is concerned with the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a singular parabolic p-biharmonic equation with logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the methods of Faedo–Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, we use the technique of modified differential inequality and improved logarithmic Sobolev inequality to obtain the global solvability, infinite and finite time blow-up phenomena, and derive the upper bound of blow-up time as well as the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena are presented.     

Solvability of the Neumann problem for a Sobolev pseudoparabolic equation

The dynamic potential constructed in this paper is used to analyze the existence of a classical solution to the Neumann problem for a Sobolev equation.     

Uniqueness and radial symmetry of least energy solution for a semilinear Neumann problem

Consider the following Neumann problem
d△u- u ＋ k（x）u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,
where d 〉 0, B1 is the unit ball in R^N, k（x） = k（｜x｜） ≠ 0 is nonnegative and in C（-↑B1）, 1 〈 p 〈 N＋2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem （＊） has no nonconstant radially symmetric least energy solution if k（x） ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that （＊） has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k（x） ≡ 1 and nonconstant if k（x） ≠ 1. In particular, for k（x） ≡ 1, do can be expressed explicitly.     

Multiple positive solutions of nonlinear Neumann problems with time and space singularities

Allowing the nonlinear term to be singular with respect to both the time and space variables, we consider the positive solutions of a nonlinear Neumann boundary value problem. By constructing two height functions and estimating the integrations of these height functions, the existence and multiplicity of positive solutions are established.     

The Neumann problem for the Laplace equation on general domains

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set G in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on ∂G. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on G a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.