Nonlinear Neumann Problem Research Articles

On the nonlinear Neumann problem with critical and supercritical nonlinearities

We investigate the solvability of the Neumann problem (1.1) involving a critical Sobolev exponent. In the first part of this work it is assumed that the coefficients $Q$ and $h$ are at least continuous. Moreover $Q$ is positive on $\,{\overl

A strongly nonlinear Neumann problem at resonance with restrictions on the nonlinearity just in one direction

Using topological degree techniques, we state and prove new sufficient conditions for the existence of a solution of the Neumann boundary value problem $$ (|x'|^{p-2} x')' +f(t, x)+ h(t, x) =0, \quad x'(0) = x'(1)=0, $$ when $h$ is bounded, $f$ satisfies a one-sided growth condition, $f + h$ some sign condition, and the solutions of some associated homogeneous problem are not oscillatory. A generalization of Lyapunov inequality is proved for a $p$-Laplacian equation. Similar results are given for the periodic problem.

On the nonlinear Neumann problem at resonance with critical Sobolev nonlinearity

We consider the Neumann problem for the equation $-{\mit \Delta } u-\lambda u = Q(x)|u|^{2^{*}-2}u$, $u\in H^1({\mit \Omega })$, where $Q$ is a positive and continuous coefficient on $\hskip 1.8pt\overline {\hskip -1.8pt{\mit \Omega }\hskip -.2pt}\hskip .

Multipeak solutions for a nonlinear Neumann problem in exterior domains

We prove the existence of multipeak positive solutions for an elliptic Neumann problem in exterior domains involving nearly critical Sobolev exponent.

Existence and multiplicity of solutions for a nonlinear Neumann problem

We consider a Neumann problem of the type -eΔu+F ′(u(x))=0 in an open bounded subset Ω of R n , where F is a real function which has exactly k maximum points. Using Morse theory we find that, for e suitably small, there are at least 2k nontrivial solutions of the problem and we give some qualitative information about them.

Existence and nonexistence of interior-peaked solution for a nonlinear Neumann problem

Existence and nonexistence of interior-peaked solution for a nonlinear Neumann problem

On the Solvability of Some Nonlinear Neumann Problems

Abstract In this paper we study the existence of solutions of some nonlinear Neumann problems and the main tool used is the fibering method.

The Bellman–Kalaba–Lakshmikantham Quasilinearization Method for Neumann Problems

In this paper, we develop a generalized quasilinearization method for a nonlinear Neumann problem and obtain a sequence of approximate solutions converging monotonically and quadratically to a solution of the problem.

Extremal solutions for nonlinear neumann problems

In this paper, we study a nonlinear Neumann problem. Assuming the existence of an upper and a lower solution, we prove the existence of a least and a greatest solution between them. Our approach uses the theory of operators of monotone type together with truncation and penalization techniques.

The nonlinear Neumann problem and sharp weighted Sobolev inequalities

We prove sharp inequalities in weighted Sobolev spaces. Our approach is based on the blow-up technique applied to some nonlinear Neumann problems.

On shape and multiplicity of solutions for a singularly perturbed Neumann problem

We investigate the effect of the topology of the boundary $\partial {\mit \Omega }$ and of the graph topology of the coefficient $Q$ on the number of solutions of the nonlinear Neumann problem $(1_d)$.

Extinction in Finite Time of Solutions to Degenerate Parabolic Equations with Nonlinear Boundary Conditions

This work concerns a nonlinear diffusion–absorption equation with nonlinear boundary flux. The main topic of interest is the problem of finite time extinction, i.e., the solutions vanish after a finite time. The sufficient and necessary conditions for occurrence of extinction are established. It is shown that extinction is caused by either strong absorption in the interior of the domain or fast diffusion combined with strong absorption through the boundary of the domain. Extinction results are also obtained for the mixed boundary value problem. In contrast to the nonlinear Neumann problem, the absorption on the boundary is no longer important, i.e., the occurrence of extinction in this case is completely determined by the effects of diffusion and interior absorption.

Existence and nonexistence of G -least energy solutions for a nonlinear Neumann problem with critical exponent in symmetric domains

Existence and nonexistence of G -least energy solutions for a nonlinear Neumann problem with critical exponent in symmetric domains

Existence and Nonexistence of Solutions of Nonlinear Neumann Problems

Existence theorems for nonnegative solutions to a class of nonlinear Neumann problems are proved. Nonexistence results are also discussed, depending either on absorption or on first-order terms. The proofs make use of a direct variational approach.

On a Class of Nonlinear Neumann Problems of Parabolic Type: Blow-Up of Solutions

We investigate large time behaviour of solutions for a class of nonlinear Neumann parabolic problems of indefinite type, possibly degenerate. Depending on the features of the problem, several parameters play a role to establish global boundedness or finite time blow-up of solutions. The occurrence of either situation is related with the existence of stationary solutions. Proofs make extensive use of monotonicity methods.

Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains

Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains

Nonradial solutions of nonlinear Neumann problems in radially symmetric domains

where Ω ⊂ R is a bounded domain with smooth boundary, λ > 0 and p > 1 are constants and ν is the unit outer normal to ∂Ω. Concerning the existence, multiplicity, and qualitative properties of solutions of (1.1)λ many interesting results have appeared; especially after Ni and Takagi ([NT1]) first discovered the spike-layer structure on the shape of least energy solutions for the subcritical problems, a lot of work has been devoted to the study of qualitative properties of solutions of (1.1)λ. For more references, we refer to [NT2] and [Wz5], in which both the subcritical exponent case (i.e. 1 < p < N+2 N−2 ) and the critical exponent case (i.e. p = N+2 N−2 ) are surveyed. In this paper, we shall focus on the case where Ω is a spherically symmetric domain, especially on the case where Ω is a ball domain. We are mainly interested in the existence and the shape of nonradial solutions of (1.1)λ. When we replace the Neumann boundary condition by the Dirichlet boundary condition the well known Gidas-Ni-Nirenberg result ([GNN]) asserts that any positive solutions must be radially symmetric. However, we shall see that contrary to its Dirichlet counterpart, (1.1)λ possesses many nonradial solutions when Ω is a ball domain. In [Wz6], we have presented an approach to this problem to construct multi-peaked solutions for (1.1)λ with the critical Sobolev exponent when Ω is a symmetric domain. We

High-energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponents

We establish the existence of positive solutions with two peaks being located on the boundary of the domain for the problem−Δu + λu = upin antipodal invariant domains including ball domains with Neumann boundary conditions. Herepis the critical Sobolev exponent (N+ 2)/(N− 2). The shape of the solutions and the location of the peaks are also studied.

The nonlinear boundary value problems for second order elliptic system of class E 2 for multiply connected domains in the plane

In this paper, nonlinear boundary value problems for second order elliptic system class E 2 for multiply connected domains in the plane is considered. We give the integral representations of Green's functions for Dirichlet problems and Neumann problems, By using the Green's functions, we mainly discuss the solvability of nonlinear Dirichlet problems and Neumann problems and give the expression of the solution.

On multiple solutions of a nonlinear Neumann problem

Lower bounds on the number of solutions to a certain class of nonlinear Neumann boundary value problems are obtained as a function of boundary data and of the parameters involved with the nonlinear term.