Quasilinear elliptic equations on $$\mathbb{R}^N $$with infinitely many solutions

We give verifiable conditions ensuring that second order quasilinear elliptic equations on have infinitely many solutions in the Sobolev space for generic right-hand sides. This amounts to translating in concrete terms the more elusive hypotheses of an abstract theorem. Salient points include the proof that a key denseness property is equivalent to the existence of nontrivial solutions to an auxiliary problem, and an estimate of the size of the set of critical points of nonlinear Schrödinger operators. Conditions for the real-analyticity of Nemytskii operators are also discussed.     

A note on nonlinear fourth-order elliptic equations on \mathbb R ^N

We established the existence of weak solutions of the fourth-order elliptic equation of the form $$\begin{aligned} \Delta ^2 u -\Delta u + a(x)u = \lambda b(x) f(u) + \mu g (x, u), \qquad x \in \mathbb{R }^N, u \in H^2(\mathbb{R }^N), \end{aligned}$$ where $\lambda $ is a positive parameter, $a(x)$ and $b(x)$ are positive functions, while $f : \mathbb{R }\rightarrow \mathbb{R }$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that the problem has at least three solutions.     

Asymmetric positive solutions for a symmetric nonlinear problem in {mathbb R}^N

We study a symmetric semilinear elliptic problem in all and we prove existence of an asymmetric positive solution by using variational arguments. The corresponding problem in dimension N=2, which provides the motivation of this work, arises in Nonlinear Optics from the study of the behaviour of optical cylindrical waveguides. Received September 28, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

Three radial positive solutions for semilinear elliptic problems in $\mathbb{R}^N}

This paper is concerned with the semilinear elliptic problem $$ \left\{ \begin{aligned} &-\Delta u=\lambda h(|x|)f(u) \ \ \ \ \ \ \ \ \ \ \text{in}\ \mathbb{R}^N, \\~ & u(x)>0\hskip 3cm \ \text{in}\ \mathbb{R}^N, \\~ &u\to 0 \hskip 3cm \ \ \ \ \text{as}\ |x|\to \infty, \end{aligned} \right. $$ where $\lambda$ is a real parameter and $h$ is a weight function which is positive. We show the existence of three radial positive solutions under suitable conditions on the nonlinearity. Proofs are mainly based on the bifurcation technique.     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

Branches of solutions to semilinear elliptic equations on mathbb{R}^N

For a large class of functions f, we consider the nonlinear elliptic eigenvalue problem We describe the behaviour of the branch of solutions emanating from an eigenvalue of odd multiplicity below the essential spectrum of the linearized problem. A sharper result is obtained in the case of the lowest eigenvalue. The discussion is based on the degree theory for proper Fredholm maps developed by P.M Fitzpatrick, J. Pejsachowicz and P.J. Rabier. Received November 13, 1996; in final form March 24, 1997     

Rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^N with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.     

A minimum principle for the problem of St-Venant in {\mathbb{R}^N, N \geq 2}

This paper deals mainly with the St-Venant problem in a convex domain ?? of ${\mathbb{R}^N, N \geq 2}$ . A minimum principle for a combination of the stress function ${\psi}$ and ${|\nabla \psi|}$ is derived. Some possible applications are indicated.     

Multibubble analysis on N-Laplace equation in {\mathbb{R}^N}

In this note, we describe the asymptotic behavior of sequences of solutions to N-Laplace equations with critical exponential growth in smooth bounded domain in ${\mathbb{R}^N}$ . Precisely we prove multibubble phenomena and obtain an energy inequality for those concentrating solutions. In fact we partly extend the corresponding two-dimensional results of Adimurthi and Struwe (J Funct Anal 175:125?C167, 2000) and Druet (Duke Math J 132:217?C269, 2006) to high dimensional case.     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.     

A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}

We study the Cauchy problem in \mathbbR^N{\mathbb{R}^N} for the parabolic equation

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

Global bifurcation for quasilinear elliptic equations on $\mathbb{R}^{N}$

In this paper we discuss the global behaviour of some connected sets of solutions of a broad class of second order quasilinear elliptic equations for where is a real parameter and the function u is required to satisfy the condition The basic tool is the degree for proper Fredholm maps of index zero in the form due to Fitzpatrick, Pejsachowicz and Rabier. To use this degree the problem must be expressed in the form where J is an interval, X and Y are Banach spaces and F is a map which is Fredholm and proper on closed bounded subsets. We use the usual spaces and . Then the main difficulty involves finding general conditions on and b which ensure the properness of F. Our approach to this is based on some recent work where, under the assumption that and b are asymptotically periodic in x as $\left| x\right| \rightarrow\infty$, we have obtained simple conditions which are necessary and sufficient for to be Fredholm and proper on closed bounded subsets of X. In particular, the nonexistence of nonzero solutions in X of the asymptotic problem plays a crucial role in this issue. Our results establish the bifurcation of global branches of solutions for the general problem. Various special cases are also discussed. Even for semilinear equations of the form our results cover situations outside the scope of other methods in the literature. Received March 30, 1999; in final form January 17, 2000 / Published online February 5, 2001     

Some pinching theorems for minimal submanifolds in \mathbb{S}^m (1) \times \mathbb{R}

Let Mn be an n-dimensional compact minimal submanifolds in Sm(1)×R.We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively.In fact,we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.