Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let Ω be a smooth bounded domain in , with N?5, a>0, α?0 and . We show that the exponent plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem

     

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

In this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary ∂Ω and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1).     

The critical Neumann problem for semilinear elliptic equations with concave perturbations

We consider the semilinear Neumann problem involving the critical Sobolev exponent with an indefinite weight function and a concave purturbation. We prove the existence of two distinct solutions.      

Positive Solutions to a Neumann Problem of Semilinear Elliptic System with Critical Nonlinearity

In this paper, we consider the Neumann boundary value problem for a system of two elliptic equations involving the critical Sobolev exponents. By means of blowing-up method, we obtain behavior of positives with low energy and asymptotic behavior of positive solutions with minimum energy as the parameters λ,μ→∞.     

On the Neumann problem with the Hardy–Sobolev potential

In this paper we investigate the solvability of the nonlinear Neumann problem (1.1) with indefinite weight functions and a critical Hardy–Sobolev nonlinearity. We examine the common effect of the shape of the graph of a weight function and the mean curvature of the boundary on the existence of solutions of problem (1.1). We also investigate the regularity of solutions.      

Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition

We study the existence of positive solutions of a linear elliptic equation with critical Sobolev exponent in a nonlinear Neumann boundary condition. We prove a result which is similar to a classical result of Brezis and Nirenberg who considered a corresponding problem with nonlinearity in the equation. Our proof of the fact that the dimension three is critical uses a new Pohoaev-type identity.AMS Subject Classification: Primary: 35J65; Secondary: 35B33.     

On the Neumann problem for elliptic equations involving the p-Laplacian

The aim of this paper is to establish the existence of an unbounded sequence of weak solutions to a Neumann problem for elliptic equations involving the p-Laplacian.     

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.     

Remarks on a multiplicity theorem for a perturbed Neumann problem

The aim of this article is to prove that, given two potential functionals Ψ₁, Ψ₂ on W1,2(Ω) which coincide on a set of the type , then under suitable summability conditions, certain local minima of Ψ₁ are local minima for Ψ₂ as well. An application of this result allows us to obtain a multiplicity theorem for a Neumann problem where we impose a less restrictive oscillating behavior on the nonlinearity than the one required in an analogous result recently established by B. Ricceri.     

Condensation of Least-energy Solutions of a Semilinear Neumann Problem

This paper is devoted to the study of tho least-energy solutions of a singularly perturbed Neumann problem involving critical Sobolev exponents. The condensation rate is given when n > 4 apd an asymptotic behavior result is obtained.     

On the structure of solutions to a class of quasilinear elliptic Neumann problems

We study the structure of positive solutions to the equation ?^mΔ_mu-u^m-1+f(u)=0 with homogeneous Neumann boundary condition. First, we show the existence of a mountain-pass solution and find that as ?→0⁺ the mountain-pass solution develops into a spike-layer solution. Second, we prove that there is an uniform upper bound independent of ? for any positive solution to our problem. We also present a Harnack-type inequality for the positive solutions. Finally, we show that if 1<m?2 holds and ? is sufficiently large, any positive solution must be a constant.     

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.     

Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Ljusternik-Schnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.     

Semilinear dirichlet problem with nearly critical exponent, asymptotic location of hot spots

We study asymptotic properties of the positive solutions of

Multiplicity results for a Neumann problem involving the p-Laplacian

In this paper, we establish some multiplicity results for the following Neumann problem: