Principle Of Symmetric Criticality Research Articles

On a planar equation involving [formula omitted]-Laplacian with zero mass and Trudinger–Moser nonlinearity

In this work, we study existence of positive solutions to a class of (2,q)-equations in the zero mass case in R2. We establish a weighted Sobolev embedding and we introduce a new Trudinger–Moser type inequality. Moreover, since we work on a suitable radial Sobolev space, we prove an appropriate version of the well-known Symmetric Criticality Principle by Palais. Finally, we study regularity of solutions applying Moser iteration scheme.

Generalized noncooperative Schrödinger–Kirchhoff–type systems in RN$\mathbb {R}^N$

AbstractWe consider a class of noncooperative Schrödinger–Kirchhof–type system, which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration–compactness principle for weighted‐variable exponent Sobolev spaces and the principle of symmetric criticality of Krawcewicz and Marzantowicz.

Radial and nonradial solutions for nonautonomous Kirchhoff problems

In this paper, we study the following nonautonomous Kirchhoff problem: where , , , is a positive constant, is a parameter, and the potential functions belong to . The existence of radially symmetric and positive solution to the above problem is first established for all when are radially symmetric and , and the range of can be extended to with the aid of a coercive type assumption on . Moreover, we show the existence of infinitely many solutions with high energies via the fountain theorem under more general assumption on which allows it to be sign‐changing. When and , we show that the above problem possesses infinitely many solutions with negative critical values for small provided that the function belongs to a suitable space. In particular, by imposing a hypothesis on the potential controlling its growth at infinity, we obtain a nonradial solution via the mountain pass theorem and the principle of symmetric criticality.

Quasilinear equation with critical exponential growth in the zero mass case

In this work we study the existence of solutions for the following class of quasilinear elliptic equations −divA(|x|)|∇u|N−2∇u=Q(|x|)f(u),inRN,where N≥2 and f has critical exponential growth. We establish conditions on the non-homogeneous weights A and Q to introduce a suitable function space where we are able to apply Variational Methods to obtain weak solutions. The main key is a Hardy type inequality for radial functions. Our approach is based on a new Trudinger–Moser type inequality, a version of the Symmetric Criticality Principle and Mountain Pass Theorem.

Yamabe problem in the presence of singular Riemannian Foliations

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

Infinitely many solutions for fractional elliptic systems involving critical nonlinearities and Hardy potentials

This paper is dedicated to studying the existence of multiple nontrivial solutions for a class of singular fractional elliptic systems involving critical nonlinearities and Hardy potentials in RN. Based upon the Caffarelli–Silvestre extension method and the Krasnoselskii genus theory, together with the symmetric criticality principle of Palais, we establish several existence results of infinitely many solutions under certain appropriate hypotheses on the weights and the parameters.

Existence of smooth solutions for a class of Euclidean bosonic equations

We study a class of non-linear equations on Euclidean space building on previous works by the present authors. These equations emerge in the study of cosmology and string theory, see for instance Calcagni, Montobbio and Nardelli (2007) [3]; Calcagni, Montobbio and Nardelli (2008) [4] and they depend on operators of the form exp⁡(−cΔ), in which Δ is the Euclidean Laplace operator and c>0. We introduce an appropriate space of functions on which this operator is well defined; this domain is continuously embedded into the scale of Sobolev spaces, and this fact allows us to investigate weak and strong solutions. We prove that, for a wide class of non-linearities, there exists non-trivial smooth solutions. Our tools are classical fixed point theorems and variational calculus. We are able to prove existence of (strong) solutions using Schaeffer's fixed point theorem, and existence of radial and non-radial non-trivial strong solutions using the variational approach and the principle of symmetric criticality.

Existence of a radial solution to a 1-Laplacian problem in [formula omitted

We obtain the existence of a radial solution to the following 1-Laplacian problem −Δ1u+u|u|=Q(x)f(u),inRN,u∈BV(RN).(0.1) The work is carried out in the space of functions of bounded variation BV(RN). The proof of the main result relies on a version of mountain pass theorem without Palais–Smale condition to Lipschitz continuous functionals, and symmetric criticality principle of Palais for non-smooth functionals .

Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials

In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in $$H^{2}\left( {\mathbb {R}}^{4}\right) $$ which are of their independent interests. (See Theorems 2.1 and 2.2.) Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form $$\begin{aligned} (-\Delta )^{2}u+\gamma u=f(u) \text {in} {\mathbb {R}}^{4} \end{aligned}$$ and the range of $$\gamma \in {\mathbb {R}}^{+}$$ , where $$f\left( s\right) $$ is the general nonlinear term having the critical exponential growth at infinity. (See Theorem 2.7.) Though the existence of the nontrivial solutions for the bi-harmonic equation with the critical exponential growth has been studied in the literature, it seems that nothing is known so far about the existence of the ground-state solutions for this class of equations involving the trapping potential introduced by Rabinowitz (Z Angew Math Phys 43:27–42, 1992). Since the trapping potential is not necessarily symmetric, classical radial method cannot be applied to solve this problem. In order to overcome this difficulty, we first establish the existence of the ground-state solutions for the equation 0.1 $$\begin{aligned} (-\Delta )^{2}u+V(x)u=\lambda s\exp (2|s|^{2})) \text {in} {\mathbb {R}}^{4}, \end{aligned}$$ when V(x) is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the Eq. (2.5) when V(x) is the Rabinowitz type trapping potential, namely it satisfies $$\begin{aligned} 0<\inf _{x \in {\mathbb {R}}^{4}} V(x)<\sup _{x \in {\mathbb {R}}^{4}} V(x)=\lim _{|x| \rightarrow +\infty } V(x). \end{aligned}$$ (See Theorem 2.8.) The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in $${\mathbb {R}}^2$$ . (See Theorem 2.9.)

On nonlinear Schrödinger equations on the hyperbolic space

We study existence of weak solutions for certain classes of nonlinear Schrödinger equations on the Poincaré ball model BN, N≥3. By using the Palais principle of symmetric criticality and suitable group theoretical arguments, we establish the existence of a nontrivial (weak) solution.

Critical point theory on convex subsets with applications in differential equations and analysis

We shall establish a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space. This variational principle has many applications in partial differential equations while unifying and generalizing several results in nonlinear analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality. As a consequence, several substantial new results are emerged. We shall also provide concrete applications in local and nonlocal partial differential equations, including the symmetry properties of the Allen-Cahn equation on bounded domains, for which the standard methodologies have major limitations to be applied.

The Lane-Emden equation with variable double-phase and multiple regime

We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical, and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality.

A non-homogeneous elliptic problem in low dimensions with three symmetric solutions

We investigate the existence of multiple solutions for a low-dimensional Neumann problem with non-standard growth set on a strip-like domain of Rn. Variational techniques, depending in particular on a minimax theorem and a version of the principle of symmetric criticality, are employed to establish the existence of three non-zero solutions endowed with axial symmetry. Some consequences are also derived.

Gradient-Type Systems on Unbounded Domains of the Heisenberg Group

The purpose of this paper is to study the existence of weak solutions for some classes of one-parameter subelliptic gradient-type systems involving a Sobolev-Hardy potential defined on an unbounded domain $\Omega_\psi$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$ ($n\geq 1$) whose geometrical profile is determined by two real positive functions $\psi_1$ and $\psi_2$ that are bounded on bounded sets. The treated problems have a variational structure and thanks to this, we are able to prove the existence of an open interval $\Lambda\subset (0,\infty)$ such that, for every parameter $\lambda\in \Lambda$, the system has at least two nontrivial symmetric weak solutions that are uniformly bounded with respect to the Sobolev $HW^{1,2}_0$-norm. Moreover, the existence is stable under certain small subcritical perturbations of the nonlinear term. The main proof, crucially based on the Palais principle of symmetric criticality, is obtained by developing a group-theoretical procedure on the unitary group $\mathbb{U}(n)=U(n)\times\{1\}$ and by exploiting some compactness embedding results into Lebesgue spaces, recently proved for suitable $\mathbb{U}(n)$-invariant subspaces of the Folland-Stein space $HW^{1,2}_0(\Omega_\psi)$. A key ingredient for our variational approach is a very general min-max argument valid for sufficiently smooth functionals defined on reflexive Banach spaces.

A compact embedding result for anisotropic Sobolev spaces associated to a strip-like domain and some applications

Let m≥1 and d≥2 be integers and consider a strip-like domain O×Rd, where O⊂Rm is a bounded Euclidean domain with smooth boundary. Furthermore, let p:O¯×Rd→R be a uniformly continuous and cylindrically symmetric function. We prove that the subspace of W1,p(x,y)(O×Rd) consisting of the cylindrically symmetric functions is compactly embedded into L∞(O×Rd) provided thatm+d<p−:=inf(x,y)∈O¯×Rd⁡p(x,y)≤p+:=sup(x,y)∈O¯×Rd⁡p(x,y)<+∞. As an application, we study a Neumann problem involving the p(x,y)-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many cylindrically symmetric weak solutions. Our approach is based on variational and topological methods in addition to the principle of symmetric criticality.

Some hemivariational inequalities in the Euclidean space

AbstractThe purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd(d≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal groupO(d) and their actions on the Sobolev spaceH1(ℝd). Moreover, under an additional hypotheses on the dimensiondand in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

Quasilinear Schrödinger equations with a positive parameter and involving unbounded or decaying potentials

We establish the existence of nonnegative and nonzero solutions for the following class of quasilinear Schrödinger equations: where , κ is a positive parameter, V and Q are potentials that can be singular at the origin, unbounded or vanishing at infinity and the nonlinearity is superlinear at infinity. In order to prove our main result, we have applied minimax methods together with careful -estimates and we need to use an argument of symmetric criticality principle type.

A group-theoretical approach for nonlinear Schrödinger equations

Abstract The purpose of this paper is to study the existence of weak solutions for some classes of Schrödinger equations defined on the Euclidean space ℝ d {\mathbb{R}^{d}} ( d ≥ 3 {d\geq 3} ). These equations have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using the Palais principle of symmetric criticality and a group-theoretical approach used on a suitable closed subgroup of the orthogonal group O ⁢ ( d ) {O(d)} . In addition, if the nonlinear term is odd, and d &gt; 3 {d&gt;3} , the existence of ( - 1 ) d + [ d - 3 2 ] {(-1)^{d}+[\frac{d-3}{2}]} pairs of sign-changing solutions has been proved. To make the nonlinear setting work, a certain summability of the L ∞ {L^{\infty}} -positive and radially symmetric potential term W governing the Schrödinger equations is requested. A concrete example of an application is pointed out. Finally, we emphasize that the method adopted here should be applied for a wider class of energies largely studied in the current literature also in non-Euclidean setting as, for instance, concave-convex nonlinearities on Cartan–Hadamard manifolds with poles.

Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4

Abstract In this paper, we establish a sharp concentration-compactness principle associated with the singular Adams inequality on the second-order Sobolev spaces in ℝ 4 {\mathbb{R}^{4}} . We also give a new Sobolev compact embedding which states W 2 , 2 ⁢ ( ℝ 4 ) {W^{2,2}(\mathbb{R}^{4})} is compactly embedded into L p ⁢ ( ℝ 4 , | x | - β ⁢ d ⁢ x ) {L^{p}(\mathbb{R}^{4},|x|^{-\beta}\,dx)} for p ≥ 2 {p\geq 2} and 0 &lt; β &lt; 4 {0&lt;\beta&lt;4} . As applications, we establish the existence of ground state solutions to the following bi-Laplacian equation with critical nonlinearity: Δ 2 ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) | x | β in ⁢ ℝ 4 , \displaystyle\Delta^{2}u+V(x)u=\frac{f(x,u)}{|x|^{\beta}}\quad\mbox{in }% \mathbb{R}^{4}, where V ⁢ ( x ) {V(x)} has a positive lower bound and f ⁢ ( x , t ) {f(x,t)} behaves like exp ⁡ ( α ⁢ | t | 2 ) {\exp(\alpha|t|^{2})} as t → + ∞ {t\to+\infty} . In the case β = 0 {\beta=0} , because of the loss of Sobolev compact embedding, we use the principle of symmetric criticality to obtain the existence of ground state solutions by assuming f ⁢ ( x , t ) {f(x,t)} and V ⁢ ( x ) {V(x)} are radial with respect to x and f ⁢ ( x , t ) = o ⁢ ( t ) {f(x,t)=o(t)} as t → 0 {t\rightarrow 0} .

Symmetric critical knots for O'Hara's energies

We prove the existence of symmetric critical torus knots for O'Hara's knot energy family Eα, α∈(2,3) using Palais' classic principle of symmetric criticality. It turns out that in every torus knot class there are at least two smooth Eα-critical knots, which supports experimental observations using numerical gradient flows.