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.

     

Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem

In this paper we study the shape of least-energy solutions to the quasilinear problem εmΔmu−um−1+f(u)=0 with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that as ε→⁰⁺, the global maximum point Pε of least-energy solutions goes to a point on the boundary ∂Ω at the rate of o(ε) and this point on the boundary approaches to a point where the mean curvature of ∂Ω achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions.     

Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, Part II

We continue our work (Y. Li, C. Zhao, Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, J. Math. Anal. Appl. 336 (2007) 1368-1383) to study the shape of least-energy solutions to the quasilinear problem ε^mΔ_mu−u^m−1+f(u)=0 with homogeneous Neumann boundary condition. In this paper we focus on the case 1<m<2 as a complement to our previous work on the case m≥2. We use an intrinsic variation method to show that as the case m≥2, when ε→0⁺, the global maximum point P_ε of least-energy solutions goes to a point on the boundary ∂Ω at a rate of o(ε) and this point on the boundary approaches a global maximum point of mean curvature of ∂Ω.     

On the existence of some new positive interior spike solutions to a semilinear Neumann problem

In this paper we are concerned with the following Neumann problem

     

Global branches of non-radially symmetric solutions to a semilinear Neumann problem in a disk

Let D⊂R² be a disk, and let f∈C³. We assume that there is a∈R such that f(a)=0 and f^′(a)>0. In this article, we are concerned with the Neumann problem

     

Multiple solutions of a class of Neumann problem for semilinear elliptic equations

The multiplicity results are obtained for solutions of the Neumann problem for nonlinear elliptic equations with unbounded nonlinearity by the Implicit Function Theorem and the Morse theory.     

The symmetry of least-energy solutions for semilinear elliptic equations

In this paper we will apply the method of rotating planes (MRP) to investigate the radial and axial symmetry of the least-energy solutions for semilinear elliptic equations on the Dirichlet and Neumann problems, respectively. MRP is a variant of the famous method of moving planes. One of our main results is to consider the least-energy solutions of the following equation:

(∗)     

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 the number of nodal solutions to a singularly perturbed Neumann problem

We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ² and f grows superlinearly. (A typical f(u) is f(u)= a₁ u₊^p – a₁ u_-^p, a₁, a₂ >0, p, q>1.) More precisely, for any positive integer K, there exists ε_K>0 such that for 0<ε<ε_K, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.     

Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory

We study a perturbed semilinear problem with Neumann boundary condition where is a bounded smooth domain of , , , if or if and is the unit outward normal at the boundary of . We show that for any fixed positive integer K any “suitable” critical point of the function generates a family of multiple interior spike solutions, whose local maximum points tend to as tends to zero. Received March 7, 1999 / Accepted October 1, 1999 / Published online April 6, 2000     

Positive solutions to a singular Neumann problem

We show the existence of positive solution for the following class of singular Neumann problem in BR with ∂u/∂ν=0 on ∂BR, where R>0, λ>0 is a positive parameter, β>0, p∈[0,1), BR=BR(0)⊂RN, and are radially symmetric nonnegative C¹ functions. Using variational methods and sub- and supersolutions, we obtain a solution for an approximated problem involving mixed boundary conditions. The limit of the approximated solutions, is a positive solution.     

Existence and multiplicity for solutions of Neumann problem for semilinear elliptic equations

The existence and multiplicity results are obtained for solutions of Neumann problem for semilinear elliptic equations by the least action principle and the minimax methods respectively.     

On the behavior of solutions to the Neumann problem in unbounded domains

The paper deals with the uniformly elliptic equation (a^ij(x)u_xi)_xj=f(x) in an unbounded domain Ω⊂ℝⁿ and its solution u(x) that satisfies the homogeneous Neumann condition. The function f has a compact support. The domain Ω has the following structure: assume that {r_m} is an increasing sequence of positive numbers, h_m=r_m+1−r_m, and the ratio h_m+1/h_m lies between positive constants C₁ and C₂. The intersection of Ω with the spherical layer between the spheres of radius r_m and r_m+1 with center at the origin satisfies a certain inequality of isoperimetric type. It is shown in this paper that the set of solutions splits up into three classes: (i) the solutions for which and ; moreover, it is shown that these limits are attained with nearly the same speed (if C₁/C₂=1, then the speed is not less than the exponential one); (ii) the solutions for each of which a constant C exists such that and u(x)−C changes its sign for large |x|; here, the convergence to C is rapid (for C₁/C₂=1 this convergence is not slower than the exponential one); (iii) the solutions that do not change their sign for large |x| and increase or decrease to +∞ or −∞, respectively, with low speed (for C₁/C₂=1 with a linear speed) (one exception is possible here: a slow convergence to a constant). Bibliography: 7 titles. Dedicated to O.A. Oleinik This work was supported by the Soros International Science Foundation. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19. pp. 000-000, 0000.     

Multiple sign-changing solutions for a semilinear Neumann problem and the topology of the configuration space of the domain boundary

We study the existence of multiple sign-changing solutions of the problem $$-d^2 \Delta u + u =f(u)\quad {\rm in}\,\Omega,\quad\dfrac{\partial u}{\partial \nu}=0 \quad {\rm in}\,\partial \Omega,$$ where d > 0 is small enough, Ω is a domain in ${\mathbb{R}^{N}}$ (N ≥ 2) whose boundary is nonempty, compact and smooth and ${f \in C(\mathbb{R},\mathbb{R})}$ is a function satisfying a subcritical growth condition. We give lower estimates of the number of the sign-changing solutions by the category of a set related to the configuration space ${\{(x,y)\in\partial\Omega\times\partial\Omega:x \neq y\}}$ of the boundary ?Ω.     

On a problem of Neumann

A conjecture widely attributed to Neumann is that all finite non-desarguesian projective planes contain a Fano subplane. In this note, we show that any finite projective plane of even order which admits an orthogonal polarity contains many Fano subplanes. The number of planes of order less than $n$ previously known to contain a Fano subplane was $O(logn)$, whereas the number of planes of order less than $n$ that our theorem applies to is not bounded above by any polynomial in $n$.     

On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions

Summary We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded Lipschitz domainsD ^d (d3), having compact boundary, with nonlinear Neumann boundary conditions on the boundary ofD. For this we use an implicit probabilistic representation, Schauder's fixed point theorem, and a recently proved Sobolev inequality forW ^1,2(D). Special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics.Research supported in part by NSF Grants DMS 8657483 and GER 9023335This article was processed by the authors using the style filepljourlm from Springer-Verlag.