Principle Of Symmetric Criticality Research Articles

Existence and multiplicity of symmetric solutions for semilinear elliptic equations with singular potentials and critical Hardy–Sobolev exponents

This paper deals with the singular semilinear elliptic problem −div(|x|−2a∇u)=μu|x|2(1+a)+Q(x)|u|p−2u|x|bp+σh(x,u)inΩ,u=0on∂Ω, where Ω⊂RN(N≥3) is a smooth bounded domain, 0∈Ω and Ω is G-symmetric with respect to a subgroup G of O(N), 0≤a<N−22, σ≥0, 0≤μ<μ¯ with μ¯=(N−2−2a2)2, a≤b<a+1, p=p(a,b)=2NN−2(1+a−b), Q(x) is continuous and G-symmetric on Ω¯ and h:Ω×R↦R is a continuous nonlinearity of lower order satisfying some conditions. Based upon the symmetric criticality principle of Palais and variational methods, we prove several existence and multiplicity results of G-symmetric solutions under certain appropriate hypotheses on σ, Q and h.

Existence and multiplicity of symmetric solutions for a class of singular elliptic problems

This paper deals with the singular elliptic problem −div(|x|β∇u)=Q(x)|x|α|u|p(β,α)−2u+h(x,u)in Ω,u=0on ∂Ω, where Ω⊂RN(N≥3) is a smooth bounded domain, 0∈Ω and Ω is G-symmetric with respect to a subgroup G of O(N), β≤0, N+β−2>0, N+α>0, α+2>β, β≥2αp(β,α), p(β,α)=2(N+α)N+β−2, Q(x) is continuous and G-symmetric on Ω¯ and h:Ω×R↦R is a continuous nonlinearity of lower order satisfying some conditions. Based upon the symmetric criticality principle of Palais and variational methods, we obtain several existence and multiplicity results of G-symmetric solutions under some assumptions on Q and h.

Infinitely many radial solutions for the [formula omitted]-Kirchhoff-type equation with oscillatory nonlinearities in [formula omitted

In this paper we consider the p(x)-Kirchhoff-type equation in RN of the form{a(∫RN|∇u|p(x)+|u|p(x)p(x)dx)(−Δp(x)u+|u|p(x)−2u)=Q(x)f(u),u⩾0,x∈RN,u(x)→0,as |x|→+∞, where 1<p(x)<N for x∈RN, Q:RN→R+ is a radial potential, f:[0,+∞)→R is a continuous nonlinearity which oscillates near the origin or at infinity and a is allowed to be singular at zero. By means of a direct variational method and the principle of symmetric criticality for non-smooth Szulkin-type functionals, the existence of infinitely many radially symmetric solutions of the problem is established. Meanwhile, the sequence of solutions in L∞-norm tends to 0 (resp., to +∞) whenever f oscillates at the origin (resp., at infinity).

Existence of nontrivial solutions for p-Laplacian variational inclusion systems in ℝ N

The authors study the existence of nontrivial solutions to p-Laplacian variational inclusion systems $$\left\{ \begin{gathered} - \Delta _p u + \left| u \right|^{p - 2} u \in \partial _1 F\left( {u,v} \right), in \mathbb{R}^N , \hfill \\ - \Delta _p v + \left| v \right|^{p - 2} v \in \partial _2 F\left( {u,v} \right), in \mathbb{R}^N , \hfill \\ \end{gathered} \right.$$ where N ≥ 2, 2 ≤ p ≤ N and F: ℝ2 → ℝ is a locally Lipschitz function. Under some growth conditions on F, and by Mountain Pass Theorem and the principle of symmetric criticality, the existence of such solutions is guaranteed.

On Palais’ principle for non-smooth functionals

If G is a compact Lie group acting linearly on a Banach space X and f is a G -invariant function on X , we provide some new versions of the so-called Palais’ principle of symmetric criticality for f : X → R ̄ , in the framework of non-smooth critical point theory. We apply the results to a class of quasi-linear PDEs associated with invariant functionals which are merely lower semi-continuous and thus could not be treated by previous non-smooth versions of the principle in the literature.

Quantization of Midisuperspace Models.

We give a comprehensive review of the quantization of midisuperspace models. Though the main focus of the paper is on quantum aspects, we also provide an introduction to several classical points related to the definition of these models. We cover some important issues, in particular, the use of the principle of symmetric criticality as a very useful tool to obtain the required Hamiltonian formulations. Two main types of reductions are discussed: those involving metrics with two Killing vector fields and spherically-symmetric models. We also review the more general models obtained by coupling matter fields to these systems. Throughout the paper we give separate discussions for standard quantizations using geometrodynamical variables and those relying on loop-quantum-gravity-inspired methods.

Arbitrarily many solutions for a perturbed [formula omitted]-Laplacian equation involving oscillatory terms

This paper deals with a perturbed p ( x ) -Laplacian equation involving oscillatory terms: - div | ∇ u | p ( x ) - 2 ∇ u + | u | p ( x ) - 2 u = Q ( x ) f ( u ) + ε g ( u ) x ∈ R N , u ⩾ 0 , u ( x ) → 0 as | x | → ∞ . By using variational methods and the non-smooth version symmetric criticality principle, we establish (a) the unperturbed problem P 0 has infinitely many distinct solutions; (b) the number of distinct solutions for P ε becomes greater and greater whenever ε is smaller and smaller. Our results are a generalization of the case of Laplacian from Kristály to the case of p ( x ) -Laplacian.

Invariant Yang–Mills connections over non-reductive pseudo-Riemannian homogeneous spaces

We study invariant gauge fields over the 4-dimensional non-reductive pseudo-Riemannian homogeneous spaces G/K recently classified by Fels & Renner (2006). Given H compact semi-simple, classification results are obtained for principal H-bundles over G/K admitting: (1) a G-action (by bundle automorphisms) projecting to left multiplication on the base, and (2) at least one G-invariant connection. There are two cases which admit nontrivial examples of such bundles and all G-invariant connections on these bundles are Yang-Mills. The validity of the principle of symmetric criticality (PSC) is investigated in the context of the bundle of connections and is shown to fail for all but one of the Fels-Renner cases. This failure arises from degeneracy of the scalar product on pseudo-tensorial forms restricted to the space of symmetric variations of an invariant connection. In the exceptional case where PSC is valid, there is a unique G-invariant connection which is moreover universal, i.e. it is the solution of the Euler-Lagrange equations associated to any G-invariant Lagrangian on the bundle of connections. This solution is a canonical connection associated with a weaker notion of reductivity which we introduce.

Infinitely many sign-changing solutions for a Schrödinger equation in RN

In this paper, we consider a Schrödinger equation − Δ u + ( λ a ( x ) + 1 ) u = f ( u ) . Applying Principle of Symmetric Criticality and the invariant set method, under some assumptions on a and f , we obtain an unbounded sequence of radial sign-changing solutions for the above equation in R N when λ > 0 large enough. As N = 4 or N ⩾ 6 , λ > 0 given, using Fountain Theorem and the Principle of Symmetric Criticality, we prove that there exists an unbounded sequence of non-radial sign-changing solutions for the above equation in R N .

Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms

We propose a direct approach for detecting arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms. Although the method works in various frameworks, we illustrate it on the problem (P ε ) { − Δ u + u = Q ( x ) [ f ( u ) + ε g ( u ) ] , x ∈ R N , N ⩾ 2 , u ⩾ 0 , u ( x ) → 0 as | x | → ∞ , where Q : R N → R is a radial, positive potential, f : [ 0 , ∞ ) → R is a continuous nonlinearity which oscillates near the origin or at infinity and g : [ 0 , ∞ ) → R is any arbitrarily continuous function with g ( 0 ) = 0 . Our aim is to prove that: (a) the unperturbed problem ( P 0 ) , i.e. ε = 0 in (P ε ) , has infinitely many distinct solutions; (b) the number of distinct solutions for (P ε ) becomes greater and greater whenever | ε | is smaller and smaller. In fact, our method surprisingly shows that (a) and (b) are equivalent in the sense that they are deducible from each other. Various properties of the solutions are also described in L ∞ - and H 1 -norms. Our method is variational and a specific construction enforces the use of the principle of symmetric criticality for non-smooth Szulkin-type functionals.

Symmetric And Nonsymmetric Solutions For a Class of Quasilinear Schrödinger Equations

Abstract We use the mountain-pass theorem combined with the principle of symmetric criticality to establish multiplicity of solutions for the class of quasilinear elliptic equations -Δu + V(z)u - Δ(u2)u = h(u) in ℝN where N ≥ 4, the potential V : ℝN → ℝ is positive and bounded away from zero and satisfies appropriate periodic and symmetric conditions. The nonlinear term h(u) has subcritical growth and satisfies a condition of the Ambrosetti-Rabinowitz type. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

A multiplicity result for a special class of gradient-type systems with non-differentiable term

We prove the existence of multiple solutions of certain systems of hemivariational inequalities, using a recent minimax result of B. Ricceri. We apply both our main theorem and the principle of symmetric criticality to obtain multiple solutions of systems of hemivariational inequalities defined on certain Sobolev spaces.

A multiplicity result for gradient-type systems with non-differentiable term

We prove the existence of multiple solutions of certain systems of hemivariational inequalities, using a recent idea of B. Ricceri. As a consequence of our main theorem we obtain the existence of multiple solutions of Schrodinger type systems. Another application involves the principle of symmetric criticality.

Existence of static solutions of the semilinear Maxwell equations

In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field ${\bf E}=0$ and the magnetic field ${\bf H}$ does not depend on time) the semilinear Maxwell equations reduce to semilinear equation where “ $\nabla\times $ ” is the curl operator, f′ is the gradient of a smooth function $f:{\mathbb{R}}^3\to{\mathbb{R}}$ and ${\bf A}:{\mathbb{R}}^3\to{\mathbb{R}}^3$ is the gauge potential related to the magnetic field ${\bf H}$ ( ${\bf H}=\nabla\times {\bf A}$ ). The presence of the curl operator causes (1) to be a strongly degenerate elliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument. Keywords: Maxwell equations, Natural constraint, Minimizing sequence Mathematics Subject Classification (2000): 35B40, 35B45, 92C15

Periodic Solutions of Schrödinger Flow from S 3 to S 2*

This paper deals with the periodic solutions of Schrodinger flow from S 3 to S 2. It is shown that the periodic solution is related to the variation of some functional and there exist S1-invariant critical points to this functional. The proof makes use of the Principle of Symmetric Criticality of Palais.

Some Applications to Variational–Hemivariational Inequalities of the Principle of Symmetric Criticality for Motreanu–Panagiotopoulos Type Functionals

Using the principle of symmetric criticality for Motreanu---Panagiotopoulos type functionals we give some existence and multiplicity results for a class of variational---hemivariational inequalities on $$\mathbb {R}$$ L+M .

A nonsmooth principle of symmetric criticality and variational–hemivariational inequalities

In this paper we prove the principle of symmetric criticality for Motreanu–Panagiotopoulos type functionals, i.e., for convex, proper, lower semicontinuous functionals which are perturbed by a locally Lipschitz function. By means of this principle a variational–hemivariational inequality is studied on certain type of unbounded strips.

Existence of nonzero weak solutions for a class of elliptic variational inclusions systems in [formula omitted

We consider the following variational inclusions system of the form − △ u + u ∈ ∂ 1 F ( u , v ) in R N , − △ v + v ∈ ∂ 2 F ( u , v ) in R N , with u , v ∈ H 1 ( R N ) , where F : R 2 → R is a locally Lipschitz function and ∂ i F ( u , v ) ( i ∈ { 1 , 2 } ) are the partial generalized gradients in the sense of Clarke. Under various growth conditions on the nonlinearity F we study the existence of nonzero weak solutions of the above system (in the sense of hemivariational inequalities), which are critical points of an appropriate locally Lipschitz function defined on H 1 ( R N ) × H 1 ( R N ) . The main tool used in the paper is the principle of symmetric criticality for locally Lipschitz functions.

EXISTENCE OF TWO NON-TRIVIAL SOLUTIONS FOR A CLASS OF QUASILINEAR ELLIPTIC VARIATIONAL SYSTEMS ON STRIP-LIKE DOMAINS

AbstractIn this paper we study the multiplicity of solutions of the quasilinear elliptic system\begin{equation} \left. \begin{aligned} -\Delta_pu\amp=\lambda F_u(x,u,v)\amp\amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)\amp\amp\text{in }\varOmega, \\ u=v\amp=0\amp\amp\text{on }\partial\varOmega, \end{aligned} \right\} \end{equation} \tag{S$_\lambda$}where $\varOmega$ is a strip-like domain and $\lambda&gt;0$ is a parameter. Under some growth conditions on $F$, we guarantee the existence of an open interval $\varLambda\subset(0,\infty)$ such that for every $\lambda\in\varLambda$, the system (S$_\lambda$) has at least two distinct, non-trivial solutions. The proof is based on an abstract critical-point result of Ricceri and on the principle of symmetric criticality.

Multiplicity Results for an Eigenvalue Problem for Hemivariational Inequalities in Strip-Like Domains

In this paper we study the multiplicity of solutions for a class of eigenvalue problems for hemivariational inequalities in strip-like domains. The first result is based on a recent abstract theorem of Marano and Motreanu, obtaining at least three distinct, axially symmetric solutions for certain eigenvalues. In the second result, a version of the fountain theorem of Bartsch which involves the nonsmooth Cerami compactness condition, provides not only infinitely many axially symmetric solutions but also axially nonsymmetric solutions in certain dimensions. In both cases the principle of symmetric criticality for locally Lipschitz functions plays a crucial role.