Nonlinear Neumann Problem Research Articles

Existence and nonexistence of positive solutions for parametric Neumann problems with $p$-Laplacian

Using variational methods based on the critical point theory and suitable truncation and comparison techniques, we study existence, multiplicity and nonexistence of positive solutions for a parametric nonlinear Neumann problem driven by the $p$-Laplacian. Our hypotheses cover the case of nonlinearities of concave-convex type whose exponents depend on the parameter.

Solvability of a nonlinear Neumann problem for systems arising from a burglary model

We study a one dimensional version of a problem that arises in mathematical modeling for burglary of houses. Our approach uses techniques inspired by coincidence degree, see Mawhin (1979). A priori estimates are obtained through a somewhat unusual combination of estimates based upon maximum or minimum properties and on L 1 -estimates of the type introduced by Ward, see Ward (1981) in some periodic problems.

Asymptotic behavior of least energy solutions for a 2D nonlinear Neumann problem with large exponent

In this paper, we consider the following elliptic problem with the nonlinear Neumann boundary condition:(Ep){−Δu+u=0onΩ,u>0onΩ,∂u∂ν=upon∂Ω, where Ω is a smooth bounded domain in R2, ν is the outer unit normal vector to ∂Ω, and p>1 is any positive number.We study the asymptotic behavior of least energy solutions to (Ep) when the nonlinear exponent p gets large. Following the arguments of X. Ren and J.C. Wei [13,14], we show that the least energy solutions remain bounded uniformly in p, and it develops one peak on the boundary, the location of which is controlled by the Green function associated to the linear problem.

Nonlinear Neumann Problems with Constraints

Nonlinear Neumann Problems with Constraints

Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems

We consider nonlinear Neumann and Dirichlet problems driven by a nonhomogeneous differential operator and a Caratheodory reaction. Our framework incorporates $p$-Laplacian equations and equations with the $(p,q)$-differential operator and with the generalized $p$-mean curvature operator. Using variational methods, together with truncation and comparison techniques and Morse theory, we prove multiplicity theorems, producing three, five or six nontrivial smooth solutions, all with sign information.

Multiple Solutions for Nonhomogeneous Neumann Differential Inclusion Problems by thep(x)-Laplacian

A class of nonlinear Neumann problems driven by p(x)-Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality) was considered. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Pass theorem are used to prove the existence of at least two nontrivial solutions.

Resonant nonlinear Neumann problems with indefinite weight

We consider nonlinear Neumann problems driven by the p-Laplacian plus an indefinite potential. First we develop the spectral properties of such differential operators. Subsequently, using these spectral properties and variational methods based on critical point theory, truncation techniques and Morse theory, we prove existence and multiplicity theorems for resonant problems.

P-Laplace equations with singular terms and p-superlinear perturbations

We consider a nonlinear parametric Neumann problem driven by the p-Laplacian with a singular term and a Caratheodory perturbation whose primitive is p- superlinear near +∞ but needs not satisfy the Ambrosetti-Rabinowitz condition. We show that when the parameter λ>0 is small, the problem has at least two nontrivial positive smooth solutions. Our method of proof uses an approximation of the original problem, which we solve using variational techniques and then pass to the limit.

A contribution to stability theory for nonlinear Neumann problem

In this paper we study the stability of the solutions of some nonlinear Neumann problems, under perturbations of the domains in the Hausdorff complementary topology. We consider the problem $${{\left\{\begin{array}{c}-\text{ div}\;\left(a\left( x,\nabla u_{\Omega}\right)\right)=0 \;\text{in}\; \Omega \\ {a\left( x, \nabla u_{\Omega}\right) \cdot \nu=0\; \text{on}\; \partial\Omega}\end{array}\right.}}$$ where \({{\mathbf{R}^n \times \mathbf{R}^n \rightarrow \mathbf{R}^n}}\) is a Caratheodory function satisfying the standard monotonicity and growth conditions of order p, 1 < p < ∞. If Ωh is a uniformly bounded sequence of connected open sets in Rn, n ≥ 2, we prove that if \({{\Omega_{h}^{c} \rightarrow \Omega^{c}}}\) in the Hausdorff metric, \({|\Omega_{h}| \rightarrow |\Omega|}\) and the geodetic distances satisfy the inequality \({d_{\Omega}\left( x,y\right) \leq \liminf_{h} d_{\Omega_{h}} \left( x,y\right)}\) for every \({x, y \in \Omega,}\) then \({\nabla u_{\Omega_h} \rightarrow\nabla u_{\Omega}}\) strongly in Lp, provided that W1, ∞(Ω) is dense in the space L1, p(Ω) of all functions whose gradient belongs to Lp(Ω, Rn).

On superlinear [formula omitted]-Laplacian problems without Ambrosetti and Rabinowitz condition

In this paper, we consider the p(x)-Laplacian equations on the bounded domain. The nonlinearity is superlinear but does not satisfy the usual Ambrosetti–Rabinowitz condition near infinity, or its dual version near zero. Existence and multiplicity results are obtained via Morse theory and modified functional methods. In a sense, we expand a recent result of Gasiński and Papageorgiou [L. Gasiński, N.S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations 42 (2011) 323–354].

Existence and uniqueness of positive solutions for the Neumann p-Laplacian

We consider a nonlinear Neumann problem driven by the p-Laplacian and with a Caratheodory reaction which satisfies only a unilateral growth restriction. Using the principal eigenvalue of an eigenvalue problem involving the Neumann p-Laplacian plus an indefinite potential, we produce necessary and sufficient conditions for the existence and uniqueness of positive smooth solutions.

Singularly perturbed nonlinear Neumann problems under the conditions of Berestycki and Lions

Let Ω be a bounded domain in R N with the boundary ∂ Ω ∈ C 3 . We consider the following singularly perturbed nonlinear elliptic problem on Ω, ε 2 Δ v − v + f ( v ) = 0 , v > 0 on Ω , ∂ v ∂ ν = 0 on ∂ Ω , where ν is the exterior normal to ∂ Ω and the nonlinearity f is of subcritical growth. It has been known that under Berestycki and Lions conditions for f ∈ C 1 ( R ) and N ⩾ 3 , there exists a solution v ε of the problem which develops a spike layer near a local maximum point of the mean curvature H on ∂ Ω for small ε > 0 . In this paper, we extend the previous result for f ∈ C 0 ( R ) and N ⩾ 2 .

Multiple Solutions for Nonlinear Neumann Problems with Asymmetric Reaction, via Morse Theory

Abstract We consider a nonlinear Neumann problem driven by the p-Laplacian and with a reaction which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)- superlinear near +∞ (but need not satisfy the Ambrosetti-Rabinowitz condition) and it is (p − 1)-linear near −∞. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions.

On the solvability of discrete nonlinear Neumann problems involving the p(x)-Laplacian

Abstract In this article, we prove the existence and uniqueness of solutions for a family of discrete boundary value problems for data f which belongs to a discrete Hilbert space W. 2010 Mathematics Subject Classification: 47A75; 35B38; 35P30; 34L05; 34L30.

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.

Positive solutions for nonlinear Neumann problems with concave and convex terms

We consider a nonlinear elliptic Neumann problem driven by the p-Laplacian with a reaction that involves the combined effects of a “concave” and of a “convex” terms. The convex term (p-superlinear term) need not satisfy the Ambrosetti–Rabinowitz condition. Employing variational methods based on the critical point theory together with truncation techniques, we prove a bifurcation type theorem for the equation.

Nonlinear Neumann equations driven by a nonhomogeneous differential operator

We consider a nonlinear Neumann problem driven by a nonhomogeneous nonlinear differential operator and with a reaction which is $(p-1)$-superlinear without necessarily satisfying the Ambrosetti-Rabinowitz condition. A particular case of our differential operator is the $p$-Laplacian. By combining variational methods based on critical point theory with truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive and the other negative).

Exponentially stable equilibria to an indefinite nonlinear Neumann problem in smooth domains

We prove existence of two nonconstant exponentially stable equilibria to the heat equation supplied with a nonlinear Neumann boundary condition in any smooth n-dimensional domain (n ≥ 2), independently of its geometry. The Neumann boundary condition reflects the fact that the flux on the boundary is proportional to the product of a prescribed bistable function of the density or concentration with an indefinite weight. Such solutions are obtained via variational methods, by minimizing the corresponding energy functional on suitable invariant sets to the semiflow generated by the parabolic problem. But this is possible only if the parameter in the boundary condition is sufficiently large, otherwise we prove using the Implicit Function Theorem the uniqueness of constant equilibrium solutions. The same theorem allows us to derive isolation and smooth dependence on the parameter for nonconstant exponentially stable equilibria found.

Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator

Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator

Precise bounds for finite time blowup of solutions to very general one-space-dimensional nonlinear Neumann problems

In this paper we analyze the asymptotic finite time blowup of solutions to the heat equation with a nonlinear Neumann boundary flux in one space dimension. We perform a detailed examination of the nature of the blowup, which can occur only at the boundary, and we provide tight upper and lower bounds for the blowup rate for “arbitrary” nonlinear functions F F , subject to very mild restrictions.