Quasilinear Neumann Problem Research Articles

Regular versus singular solutions in quasilinear indefinite problems with sublinear potentials

The aim of this paper is analyzing existence, multiplicity, and regularity issues for the positive solutions of the quasilinear Neumann problem{−(u′/1+(u′)2)′=λa(x)f(u),0<x<1,u′(0)=u′(1)=0. Here, (u′/1+(u′)2)′ is the one-dimensional curvature operator, λ∈R is a parameter, which is generally taken positive, the weight a(x) changes sign, and, in most occasions, the function f(u) has a sublinear potential F(u) at ∞. Our discussion displays the manifold patterns occurring for these solutions, depending on the behavior of the potential F(u) at u=0, and, possibly, at infinity, and of the weight function a(x) at its nodal points.

Branches of positive solutions of a superlinear indefinite problem driven by the one-dimensional curvature operator

This paper aims at proving the existence and the localization of an unbounded connected set of positive regular solutions (λ,u) of the quasilinear Neumann problem −(u′/1+(u′)2)′=λa(x)f(u),0<x<1,u′(0)=u′(1)=0,bifurcating from u=0 as λ→+∞. Here, (u′/1+(u′)2)′ is the one-dimensional curvature operator, λ∈R is a parameter, the weight a changes sign, and the function f is superlinear at 0. A novel approach is introduced based on the explicit construction of non-ordered sub and supersolutions.

Existence of solution for a quasilinear elliptic Neumann problem involving multiple critical exponents

In this paper, we study the Neumann boundary value problem to a quasilinear elliptic equation with the critical Sobolev exponent and critical Hardy–Sobolev exponent, and prove the existence of nontrivial nonnegative solution by means of variational method.

Nonconstant positive radial solutions for Neumann problem involving the mean extrinsic curvature operator

Let B be the unit ball in RN with N≥2. Let f∈C1([0,∞),R), f(0)=0, f(β)=β for some β∈(0,∞), f(s)<sfors∈(0,β),f(s)>sfors∈(β,∞) and f′(β)>λkr, where λkr is the k-th radial eigenvalue of −Δ+I in the unit ball with Neumann boundary condition. We use the unilateral global bifurcation theorem to show the existence of nonconstant, positive radial solutions of the quasilinear Neumann problem−div(∇u1−|∇u|2)+u=f(u)inB,∂νu=0on∂B.

A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth

AbstractLet 1 &lt; p &lt; +∞ and let Ω ⊂ ℝN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type $-\Delta _pu = f(u),\quad u &gt; 0\,{\rm in }\,\Omega ,\quad \partial _\nu u = 0\,{\rm on }\,\partial \Omega .$We suppose that f(0) = f(1) = 0 and that f is negative between the two zeros and positive after. In case Ω is a ball, we also require that f grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focussing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behaviour (around 1) of non-constant radial solutions.

Existence of multiple solutions for a quasilinear Neumann problem with critical exponent

The main purpose of this paper is to establish the existence and multiplicity of nontrivial solutions for a quasilinear Neumann problem with critical exponent. It is shown, by the methods of the Lions concentration-compactness principle and the mountain pass lemma, that under certain conditions, the existence of nontrivial solutions are obtained.

Multiplicity of positive solutions for a critical quasilinear Neumann problem

We establish the multiplicity of positive solutions to a quasilinear Neumann problem in expanding balls and hemispheres with critical exponent in the boundary condition.

Positive solutions for nonlinear elliptic problems with dependence on the gradient

We consider a quasilinear Neumann problem with a differential operator and a reaction term, both dependent on u and Du. Using topological methods together with suitable truncation and comparison techniques, we show that the problem has at least one positive smooth solution.

Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem

We study the structure of the set of the positive regular solutions of the one-dimensional quasilinear Neumann problem involving the curvature operator −(u′/1+(u′)2)′=λa(x)f(u),u′(0)=0,u′(1)=0. Here λ∈R is a parameter, a∈L1(0,1) changes sign, and f∈C(R). We focus on the case where the slope of f at 0, f′(0), is finite and non-zero, and the potential of f is superlinear at infinity, but also the two limiting cases where f′(0)=0, or f′(0)=+∞, are discussed. We investigate, in some special configurations, the possible development of singularities and the corresponding appearance in this problem of bounded variation solutions.

Multiplicity of Solutions to the Quasilinear Neumann Problem in the 3-Dimensional Case

We consider the quasilinear Neumann problem for equation with p-Laplacian in expanding three-dimensional balls. We prove that the number of essentially different positive solutions uboundedly increases with growth of radius.

Group-invariant multiple solutions for quasilinear elliptic problems on strip-like domains

In the present paper we prove a multiplicity result for a model quasilinear Neumann problem (Nλ) on strip-like domains, depending on a positive parameter λ. By variational methods, we prove that for large values of λ, problem (Nλ) has at least two non-zero weak solutions, while there exists at least a λ˜>0 such that problem (Nλ˜) has at least three non-zero weak solutions. These solutions show symmetry properties with respect to certain group actions.

On a quasilinear system arising in the theory of superconductivity

We examine the regularity of the solution of a quasilinear system involving the curl of vector fields. This system arises in the mathematical theory of superconductivity. The C2+α regularity was obtained by Bates and Pan under the condition that Ω is simply connected and has no holes, and that the normal component of the curl of the boundary data vanishes. The aim of this paper is to remove these technical restrictions on the topology of the domain and on the boundary data. By carefully studying the related quasilinear Neumann problem, we obtain the C2+α regularity without assuming these technical conditions.

Multiple solutions for some quasilinear Neumann problems in exterior domains

In this paper, we show that if $K(x)$ satisfies suitable conditions, then the quasilinear Neumann problem $-\Delta_pu+|u|^{p-2}u=K(x)|u|^{q-2}u$ in an exterior domain $\varOmega$ has at least two solutions, of which one is a positive ground-state solution and one is a nodal solution.

Some multiplicity results for quasilinear Neumann problems

In this paper, using a recent two local minima result of B.Ricceri and the Palais-Smale property, we establish two results ensuring the existence of three solutions for the following Neumann problem: $$ \left\{ {\begin{array}{*{20}l} {{ - \Delta _{p} u + \alpha {\left( x \right)}|u|^{{p - 2}} u = \alpha {\left( x \right)}f{\left( u \right)} + \lambda g{\left( {x,\,u} \right)}} \hfill} & {{{\text{in}}\;\Omega } \hfill} \\ {{\frac{{\partial u}} {{\partial v}} = 0} \hfill} & {{{\text{on}}\;\partial \Omega } \hfill} \\ \end{array} } \right.. $$

A quasilinear Neumann problem with discontinuous nonlinearity

AbstractIn 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.

Asymptotic analysis of a class of minimization problems in a thin multidomain

We consider a quasilinear Neumann problem with exponent \(p\in ]1,+\infty[\), in a multidomain of \({\bf R}^N\), \(N\geq2\), consisting of two vertical cylinders, one placed upon the other: the first one with given height and small cross section, the other one with small height and given cross section. Assuming that the volumes of the two cylinders tend to zero with same rate, we prove that the limit problem is well posed in the union of the limit domains, with respective dimension 1 and \(N-1\). Moreover, this limit problem is coupled if \(p>N-1\) and uncoupled if \(1<p\leq N-1\).

Asymptotic analysis for monotone quasilinear problems in thin multidomains

In this paper we perform the asymptotic analysis of a class of monotone quasilinear Neumann problems, with exponent $p\in ]1,+\infty[$ and nonstandard transmission condition, originating by change of variables from the quasilinear Neumann problem in a thin multidomain. This completes the $\Gamma$-convergence approach previously considered by the same authors. In particular, the corrector type result which is given here is more general.

Asymptotics of a Solution to the Quasiliniear Neumann Problem in a Neighborhood of a Conical Point

The existence of a small solution to the quasilinear Neumann problem is established in a H\"older class with separated asymptotics. The lower-order asymptotic terms are constructed. The homogeneity exponents of these terms turn out to be dependent on the value of the solution at a conical point. Bibliography: 33 titles.

Coupled and uncoupled limits for a N -dimensional multidomain Neumann problem

We consider the quasilinear Neumann problem with exponent p, in a multi-domain of R-N made of the union of a cylinder with given height and small cross section and a cylinder with small height and ...

Multiple positive solutions of some quasilinear neumann problems

Using the fibering method we prove the existence of at least two positive solutions for a class of non-monotone quasilinear elliptic equations with nonlinear Neuman boundary conditions