Hardy Terms Research Articles

The Numbers of Positive Solutions by the Lusternik-Schnirelmann Category for a Quasilinear Elliptic System Critical with Hardy Terms

In this paper, we study the quasilinear elliptic system with Sobolev critical exponent involving both concave-convex and Hardy terms in bounded domains. By employing the technique introduced by Benci and Cerami (1991), we obtain at least cat(Ω)+1 distinct positive solutions.

Self-Esteem of Teens with Visual Impairments as a Predictor of Hardiness

The paper discusses the problem of visually impaired adolescents’ hardiness and its predictors. The results show that adolescents with visual impairments do not differ from “healthy” peers in terms of hardiness, however, their self-esteem is different from adolescents with normal vision. Predictors of hardiness are: idealized representations of one’s own health and well-being for blind adolescents; actual self-esteem of happiness and perspective self-esteem for adolescents with low vision; actual self-esteem of happiness and mirrored self-esteem for “healthy” adolescents.

Critical Schrödinger-Hardy systems in the Heisenberg group

The paper is focused on existence of nontrivial solutions of a Schrodinger-Hardy system in the Heisenberg group, involving critical nonlinearities. Existence is obtained by an application of the mountain pass theorem and the Ekeland variational principle, but there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the Hardy terms as well as critical nonlinearities.

P-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

Abstract This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

P⁢(x)p(x)-biharmonic operator involving the p⁢(x)p(x)-Hardy inequality

Abstract In this work, we investigate the spectrum denoted by Λ for the p ⁢ ( x ) {p(x)} -biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that sup ⁡ Λ = + ∞ {\sup\Lambda=+\infty} . Moreover, we prove that inf ⁡ Λ > 0 {\inf\Lambda>0} if and only if the domain Ω satisfies the p ⁢ ( x ) {p(x)} -Hardy inequality.

Schrödinger–Hardy systems involving two Laplacian operators in the Heisenberg group

In this paper we first prove existence of nontrivial nonnegative solutions of a Schrödinger–Hardy system in the Heisenberg group, driven by two possibly different Laplacian operators. The main originality of the paper is to work in the Heisenberg group. In fact several new theorems have to be proved in order to overcome the difficulties arising in the new framework, also due to the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Finally, we discuss and prove existence even for systems in the Heisenberg group, including critical nonlinear terms.

The existence of solutions for quasilinear elliptic problems with multiple Hardy terms

In this paper, we investigate the quasilinear elliptic equations involving multiple Hardy terms with Dirichlet boundary conditions on bounded smooth domains Ω ⊂ R N N ≥ 3 , and prove the multiplicity of solutions by employing Ekeland’s variational principle.

Entire solutions for critical $p$-fractional Hardy Schrödinger Kirchhoff equations

Existence theorems of nonnegative entire solutions of stationary critical p-fractional Hardy Schr¨odinger Kirchhoff equations are presented in this paper. The equations we treat deal with Hardy terms and critical nonlinearities and the main theorems extend several recent results on the topic. The paper contains also some open problems.

Solutions for the quasi-linear elliptic problems involving the critical Sobolev exponent

In this article, we study the existence and multiplicity of positive solutions for the quasi-linear elliptic problems involving critical Sobolev exponent and a Hardy term. The main tools adopted in our proofs are the concentration compactness principle and Nehari manifold.

Effects of soil temperature on bud break, shoot and needle growth, and frost hardiness in Pinus sylvestris var. mongolica saplings during dehardening

The present study evaluated the influence of delayed soil thawing on the growth and frost hardiness of Pinus sylvestris var. mongolica saplings between September 2011 and August 2012 in Northern China (41°35′N, 116°32′E). The saplings were randomly divided into three treatments: control, snow removal, and snow removal + insulation. The control saplings were grown under natural conditions. Snow cover was removed from the saplings in the snow removal treatment. For the snow removal + insulation treatment, snow cover was removed and soil surface was covered with a layer of hay (15 cm). The bud break, current-year shoot elongation and diameter as well as current-year needle elongation were measured and compared among the three treatments. The frost hardiness of previous-year stems and needles was assessed using the electrical impedance spectroscopy method. The saplings in the snow removal + insulation treatment exhibited delayed bud break, needle and shoot growth, as well as cessation of needle and shoot growth compared to the other treatments. In contrast, the growth of shoots and needles was generally higher in the snow removal treatment. A linear relationship was observed between plant growth, especially needle elongation, and temperature sum at soil depth of 20 cm. During dehardening, there were no differences in frost hardiness for both stems and needles among the treatments. In conclusion, snow removal + insulation delayed growth, whereas snow removal accelerated the growth. The frost hardiness of saplings during dehardening did not differ among treatments significantly. The possible reduction of snow cover and increase of soil freezing under warming climate would not have negative impacts on saplings in terms of frost hardiness during dehardening. Nevertheless, increased soil freezing would have unfavorable influences in terms of shoot and needle growth.

Existence of entire solutions for Schrödinger–Hardy systems involving two fractional operators

This paper deals with the existence of nontrivial nonnegative solutions of Schrödinger–Hardy systems driven by two possibly different fractional ℘-Laplacian operators, via various variational methods. The main features of the paper are the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Moreover, we consider systems including critical nonlinear terms, as treated very recently in literature, and present radial versions of the main theorems. Finally, we briefly show how to extend the previous results when the fractional Laplacian operators are replaced by more general elliptic nonlocal integro–differential operators.

Positive Radial Solutions for a Class of Semilinear Elliptic Problems Involving Critical Hardy-Sobolev Exponent and Hardy Terms

In this paper, we investigate the solvability of a class of semilinear elliptic equations which are perturbation of the problems involving critical Hardy-Sobolev exponent and Hardy singular terms. The existence of at least a positive radial solution is established for a class of semilinear elliptic problems involving critical Hardy-Sobolev exponent and Hardy terms. The main tools are variational method, critical point theory and some analysis techniques.

Qualitative properties of positive solutions of quasilinear equations with Hardy terms

Abstract In this paper, we are concerned with the following quasilinear PDE with a weight: - div ⁡ A ⁢ ( x , ∇ ⁡ u ) = | x | a ⁢ u q ⁢ ( x ) , u > 0 in ⁢ ℝ n , -\operatorname{div}A(x,\nabla u)=|x|^{a}u^{q}(x),\qquad u>0\quad\text{in }% \mathbb{R}^{n}, where n ≥ 1 {n\geq 1} , q > p - 1 {q>p-1} with p ∈ ( 1 , 2 ] {p\in(1,2]} and a ≤ 0 {a\leq 0} . The positive weak solution u of the quasilinear PDE is 𝒜 {\mathcal{A}} -superharmonic. We also consider an integral equation involving the Wolff potential u ⁢ ( x ) = R ⁢ ( x ) ⁢ W β , p ⁢ ( | y | a ⁢ u q ⁢ ( y ) ) ⁢ ( x ) , u > 0 in ⁢ ℝ n , u(x)=R(x)W_{\beta,p}(|y|^{a}u^{q}(y))(x),\qquad u>0\quad\text{in }\mathbb{R}^{% n}, which the positive solution u of the quasilinear PDE satisfies. Here β > 0 {\beta>0} and p ⁢ β < n {p\beta<n} . When - a > p ⁢ β {-a>p\beta} or 0 < q ≤ ( n + a ) ⁢ ( p - 1 ) n - p ⁢ β {0<q\leq\frac{(n+a)(p-1)}{n-p\beta}} , there does not exist any positive solution to this integral equation. On the other hand, when 0 ≤ - a < p ⁢ β {0\leq-a<p\beta} and q > ( n + a ) ⁢ ( p - 1 ) n - p ⁢ β {q>\frac{(n+a)(p-1)}{n-p\beta}} , the positive solution u of the integral equation is bounded and decays with the fast rate n - p ⁢ β p - 1 {\frac{n-p\beta}{p-1}} if and only if it is integrable (i.e., it belongs to L n ⁢ ( q - p + 1 ) p ⁢ β + a ⁢ ( ℝ n ) {L^{\frac{n(q-p+1)}{p\beta+a}}(\mathbb{R}^{n})} ). However, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one p ⁢ β + a q - p + 1 {\frac{p\beta+a}{q-p+1}} . In addition, we also discuss the case of - a = p ⁢ β {-a=p\beta} . Thus, all the properties above are still true for the quasilinear PDE.

Existence and multiplicity of positive solutions for p-Laplacian elliptic equations

We study a p-Laplacian elliptic equation with Hardy term and Hardy-Sobolev critical exponent, where the nonlinearity is $(p-1)$ -sublinear near zero and $(p^{\ast}(s)-1)$ -sublinear near infinity ( $p^{\ast}(s)=\frac{p(N-s)}{N-p}$ is the Hardy-Sobolev critical exponent). By using variational methods and some analysis techniques, we obtain the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation. To the best of our knowledge, no result has been published concerning the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation.

Nonexistence of p-Laplace equations with multiple critical Sobolev–Hardy terms

In this paper, we study the nonexistence of solutions for p -Laplace equations with critical Sobolev–Hardy terms and singular terms by using the Pohozaev identity. And the results can be generalized to the case of p -Laplace involving multiple critical Sobolev–Hardy terms and singular terms.

Solutions of the Quasilinear Elliptic Systems with Combined Critical Sobolev–Hardy Terms

We study the existence of multiple solutions for a quasilinear elliptic system. Based on the Ambrosetti–Rabinowitz mountain-pass theorem and the Rabinowitz symmetric mountain-pass theorem, we establish several existence and multiplicity results for the solutions and G-symmetric solutions under certain suitable conditions.

The Effectiveness of Group Training of CBT-Based Stress Management on Anxiety, Psychological Hardiness and General Self-Efficacy Among University Students.

Background:Admission to university is a very sensitive period of life for efficient, active, and young workforces in any country, and it is mostly associated with many changes in social and human relationships. These changes lead to anxiety in students. Moreover, humans need certain functions in order to adaptively deal with different life situations and challenges. By training stress management, these functions can help human acquire the required abilities.Objective:The present study was aimed at investigating the effectiveness of stress management training in anxiety, psychological hardiness, and general self-efficacy among university students.Method:The study was a quasi-experimental intervention (pretest-posttest-follow-up) including a control group, it was a fundamental applied study. The statistical population consisted of all students of Islamic Azad University, Karaj, Iran. Convenient sampling was employed to select 30 students who were divided into an experimental group (n=15) and a control group (n=15). Before stress management training, both groups filled out Beck Anxiety Inventory, Long and Goulet scale of psychological hardiness, and General Self-efficacy Scale (GSE-10). Afterwards, the experimental group was provided with stress management training. And after the experiment, the abovementioned questionnaires and scales were responded by the two groups. Finally the collected data were analyzed and compared using one-way MANOVA.Results:The results of MANOVA indicated that there was a significant difference between the two groups in terms of anxiety, hardiness, and general self-efficacy (p<0.001).Conclusion:According to the results of the present study and those of previous investigations that are in agreement with those of the present study, it can be concluded that stress management among university students cause anxiety to drop; moreover, it enhances their psychological hardiness and self-efficacy. In regard with the role and importance of stress management, training this skill should be included in educational plans of university.

Quasilinear elliptic equations with Hardy terms and Hardy-Sobolev critical exponents: nontrivial solutions

In this paper, we obtain one positive solution and two nontrivial solutions of a quasilinear elliptic equation with p-Laplacian, Hardy term and Hardy-Sobolev critical exponent by using variational methods and some analysis techniques. In particular, our results extend some existing ones.

Existence of multiple positive weak solutions and estimates for extremal values to a class of elliptic problems with Hardy term and singular nonlinearity

This paper is devoted to the study of a class of elliptic equations of the form:−Δu−λ|x|2u=h(x)u−q+μW(x)up,x∈Ω\\{0} with Dirichlet boundary conditions, where 0∈Ω⊂RN (N≥3) is a bounded domain with smooth boundary ∂Ω, μ>0 is a parameter, 0<λ<Λ=(N−2)24, 0<q<1<p<2⁎−1=N+2N−2, h(x)>0 and W(x) is a given function with the set {x∈Ω:W(x)>0} of positive measure. Using variational methods, we establish some existence and multiplicity of positive solutions and provide uniform estimates of extremal values for the problem.

Elliptic equations with critical nonlinearities and Hardy terms

Abstract Using variational methods, we prove the existence of positive solutions to an elliptic equation involving critical nonlinearities and multiple inverse square potentials.