Existence and multiplicity of nontrivial solutions for an elliptic equation on ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> with indefinite linear part

 Via the Linking Theorem and Pseudo-index theory, we consider the existence and multiplicity of nontrivial solutions for a class of elliptic problems in all of ℝ ^N with indefinite linear part involving resonance and non-resonance at any eigenvalue. Received: 9 September 2002 / Revised version: 14 February 2003 Published online: 24 April 2003 Mathematics Subject Classification (2000): 35J20, 35J70     

On the periodic solutions of certain class of seventh-order differential equations

The aim of this paper is to investigate sufficient conditions (Theorem 1) for the nonexistence of nontrivial periodic solutions of equation (1.1) withp ≡ 0 and (Theorem 2) for the existence of periodic solutions of equation (1.1).     

Existence and regularity properties of non-isotropic singular elliptic equations

We establish existence and sharp regularity results for solutions to singular elliptic equations of the order u ^−β , 0 < β < 1, with gradient dependence and involving a forcing term λ f(x, u). Our approach is based on a singularly perturbed technique. We show that if the forcing parameter λ > 0 is large enough, our solution is positive. For λ small solutions vanish on a nontrivial set and therefore they exhibit free boundaries. We also establish regularity results for the free boundary and study the asymptotic behavior of the problem as b\searrow 0{\beta\searrow 0} and b\nearrow 1{\beta\nearrow 1}. In the former, we show that our solutions u _β converge to a C ^1,1 function which is a solution to an obstacle type problem. When b\nearrow 1{\beta\nearrow 1} we recover the Alt-Caffarelli theory.     

Existence of multiple solutions with precise sign information for superlinear Neumann problems

We consider a nonlinear Neumann problem driven by the p-Laplacian differential operator and having a p-superlinear nonlinearity. Using truncation techniques combined with the method of upper–lower solutions and variational arguments based on critical point theory, we prove the existence of five nontrivial smooth solutions, two positive, two negative and one nodal. For the semilinear (i.e., p = 2) problem, using critical groups we produce a second nodal solution. This paper was completed while N.S. Papageorgiou was visiting the University of Aveiro as an invited scientist. The hospitality and financial support of the host institution are gratefully acknowledged. V. Staicu acknowledges partial financial support from the Portuguese Foundation for Sciences and Technology (FCT) under the project POCI/MAT/55524/2004.     

Existence and multiplicity of solutions for asymptotically linear, noncoercive elliptic equations

First we prove the existence of a nontrivial smooth solution for a p-Laplacian equation with a (p − 1)-linear nonlinearity and a noncoercive Euler functional, under hypotheses including resonant problems with respect to the principal eigenvalue of (-D_p, W^1,p₀(Z)){(-{\it \Delta}_p,\,W^{1,p}_0(Z))} . Then, for the semilinear problem (i.e., p = 2), assuming nonuniform nonresonance at infinity and zero, we prove a multiplicity theorem which provides the existence of at least three nontrivial solutions, two being of opposite constant sign. Our approach combines minimax techniques with Morse theory and truncation arguments.     

On a nonlinear elliptic problem with critical potential in ?2

Consider the existence of nontrivial solutions of homogeneous Dirichlet problem for a nonlinear elliptic equation with the critical potential in ℝ². By establishing a weighted inequality with the best constant, determine the critical potential in ℝ², and study the eigenvalues of Laplace equation with the critical potential. By the Pohozaev identity of a solution with a singular point and the Cauchy-Kovalevskaya theorem, obtain the nonexistence result of solutions with singular points to the nonlinear elliptic equation. Moreover, for the same problem, the existence results of multiple solutions are proved by the mountain pass theorem.     

Existence and stability of nontrivial periodic solutions of periodically forced discrete dynamical Systems

The local existence and local asymptotic stability of nontrivial p-periodic solutions of p-periodically forced discrete systems are proven using Liapunov-Schmidt methods. The periodic solutions bifurcate transcritically from the trivial solution at the critical value n=n^cr of the bifurcation parameter with a typical exchange of stability. If the trivial solution loses (gains) stability as n is increased through n^cr , then the periodic solutions on the nontrivial bifurcating branch are locally asymptotically stable if and only if they correspond to n>n^cr (n n^cr ).     

Two Positive Solutions of a Quasilinear Elliptic Dirichlet Problem

For a class of second order quasilinear elliptic equations we establish the existence of two non–negative weak solutions of the Dirichlet problem on a bounded domain, Ω. Solutions of the boundary value problem are critical points of C ¹–functional on H₀¹(W){H_0^1(\Omega)}. One solution is a local minimum and the other is of mountain pass type.     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

Symmetry breaking for a class of semilinear elliptic problems and the bifurcation diagram for a 1-dimensional problem

We study the behaviour of the positive solutions to the Dirichlet problem IR ⁿ in the unit ball in IR ^R wherep<(N+2)/(N−2) ifN≥3 and λ varies over IR. For a special class of functionsg viz.,g(x)=u ₀ ^p (x) whereu ₀ is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric positive solutions. WhenN=1, we obtain the complete bifurcation diagram for the positive solution curve.     

The mountain pass method in the problem on a nontrivial solution of a quasilinear equation with a parameter in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In the present paper, we consider a quasilinear elliptic equation in ℝ ^N with a parameter whose values lie in a neighborhood of an eigenvalue of the linear problem. To prove the existence of a nontrivial solution, we use a modification of the conditional mountain pass method. The difficulties related to the lack of compactness of the Sobolev operator in the case of an unbounded domain are eliminated with the use of the Lions concentration-compactness method.     

Nonlinear elliptic equations with large supercritical exponents

The paper deals with the existence of positive solutions of the problem -Δ u=u^p in Ω, u=0 on ∂Ω, where Ω is a bounded domain of , n≥ 3, and p>2. We describe new concentration phenomena, which arise as p→ +∞ and can be exploited in order to construct, for p large enough, positive solutions that concentrate, as p→ +∞, near submanifolds of codimension 2. In this paper we consider, in particular, domains with axial symmetry and obtain positive solutions concentrating near (n-2)-dimensional spheres, which approach the boundary of Ω as p→ +∞. The existence and multiplicity results we state allow us to find positive solutions, for large p, also in domains which can be contractible and even arbitrarily close to starshaped domains (while no solution can exist if Ω is starshaped and , as a consequence of the Pohožaev's identity). Mathematics Subject Classification (2000) 35J20, 35J60, 35J65     

Existence and multiplicity results for the nonlinear Schrödinger–Maxwell systems

In this paper, we study the nonlinear Schrödinger–Maxwell system where the potential V and the primitive of g are allowed to be sign‐changing, and g is local superlinear. Under some simple assumptions on V,Q and g, we establish some existence criteria to guarantee that the aforementioned system has at least one nontrivial solution or infinitely many nontrivial solutions by using critical point theory. Recent results in the literature are generalized and significantly improved. Copyright © 2015 John Wiley & Sons, Ltd.     

EXISTENCE OF PERIODIC SOLUTIONS FOR A DIFFERENTIAL INCLUSION SYSTEMS INVOLVING THE p（t）-LAPLACIAN

We study a nonlinear periodic problem driven by the p(t)-Laplacian and having a nonsmooth potential (hemivariational inequalities). Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution under the generalized superquadratic.     

The twistor theory of the Hermitian Hurwitz pair (ℂ4(I 2,2),ℝ(I 2,3))

An analogue of the twistor theory is given for the Hermitian Hurwitz pair(ℂ⁴(I _2,2),ℝ(I _2,3)). In Sect. 2 a concept of Hurwitz twistors is introduced and a counterpart of the Penrose correspondence is obtained. It is proved that there exists a one-to-one correspondence between the twistors on the (1,3)-space and the (2,2)-space, which is called the duality theorem for Hurwitz twistors (Theorem 1). In Sect. 3. a concept of spinor equations is introduced for an Hermitian Hurwitz pair (abbreviated as HHP) and the duality theorem for solutions of the spinor equations is proved (Theorem 2). In Sect. 4 we give an elementary proof of the Penrose theory on the base of our Key Lemma. Then we can give the desired correspondence explicitly. In sect. 5 we consider the Penrose theory in the context of HHPs. At first we give a local version. It is proved that every solution of the spinor equation on the (2,2)-space can be represented as a ∂-harmonic one-form. By use of this result, we can get a direct relationship between the complex analysis and spinor theory on some open setM ⁺, which is called as “semi-global version” of the Penrose theory (Theorem 7). Moreover, we can get the original Penrose theory by use of the Penrose transformation (Theorem 5). Research of the first author partially supported by the State Committee for Scientific Research (KBN) grant PB 2 P03A 016 10 (Sections 1, 3 and 5 of the paper), and partially by the grant of the University of Łódź no. 505/485 (sections 2 and 4).     

Positive solutions and multiple solutions for periodic problems driven by scalar p ‐Laplacian

In this paper we study a nonlinear second order periodic problem driven by a scalar p ‐Laplacian and with a nonsmooth, locally Lipschitz potential function. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of nontrivial positive solutions and then establish the existence of a second distinct solution (multiplicity theorem) by strengthening further the hypotheses. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The global attractor of a non-local PDE model with delay for population dynamics in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we consider a non-local PDE model with delay for population dynamics in ℝ ⁿ . First, we prove the existence and uniqueness of weak solutions under some suitable decayed assumptions on non-local term at infinity. Then, we obtain the global attractor by proving ω-limit compactness property of the solution operator semigroup.     

Multiple positive solutions of an elliptic equation with a convex-concave nonlinearity containing a sign-changing term

We study the existence of multiple positive solutions to a nonlinear Dirichlet problem for the p-Laplacian (in a bounded domain in ℝ ^N ) with a concave nonlinearity and with a nonlinear perturbation involving a function of the spatial variable whose sign can change the character of concavity. Under two different sets of conditions imposed on the perturbation, we prove the existence of two and three positive solutions, respectively.