scholarly journals Existence and multiplicity results for quasilinear equations in the Heisenberg group

2019 ◽  
Vol 39 (2) ◽  
pp. 247-257 ◽  
Author(s):  
Patrizia Pucci
Keyword(s):  
Heisenberg Group ◽  
Quasilinear Equation ◽  
Divergence Form ◽  
Infinitely Many Solutions ◽  
Entire Solutions ◽  
Quasilinear Equations ◽  
Multiplicity Results ◽  
Minimax Theory ◽  
Existence And Multiplicity ◽  
General Elliptic

In this paper we complete the study started in [Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl. 4 (2019)] on entire solutions for a quasilinear equation \((\mathcal{E}_{\lambda})\) in \(\mathbb{H}^{n}\), depending on a real parameter \(\lambda\), which involves a general elliptic operator \(\mathbf{A}\) in divergence form and two main nonlinearities. Here, in the so called sublinear case, we prove existence for all \(\lambda\gt 0\) and, for special elliptic operators \(\mathbf{A}\), existence of infinitely many solutions \((u_k)_k\).

scholarly journals A Hopf-type Boundary Point Lemma for Pairs of Solutions to Quasilinear Equations

2019 ◽  
Vol 62 (3) ◽  
pp. 607-621 ◽  
Author(s):  
Leobardo Rosales
Keyword(s):  
Weak Solutions ◽  
Boundary Point ◽  
Quasilinear Equation ◽  
Divergence Form ◽  
Quasilinear Equations ◽  
Differentiable Functions ◽  
The Difference ◽  
Type Boundary

AbstractWe present a Hopf boundary point lemma for the difference between two Hölder continuously differentiable functions, each weak solutions to a divergence-form quasilinear equation, under mild boundedness assumptions on the coefficients of this equation.

scholarly journals Multiple solutions for Grushin operator without odd nonlinearity

2019 ◽  
Vol 13 (07) ◽  
pp. 2050131 ◽  
Author(s):  
Mohamed Karim Hamdani
Keyword(s):  
Mountain Pass Theorem ◽  
Infinitely Many Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Grushin Operator ◽  
Clark’S Theorem ◽  
Superlinear Growth ◽  
Degenerate Operator ◽  
Strongly Degenerate ◽  
Odd Nonlinearity

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: [Formula: see text] and [Formula: see text] where [Formula: see text] is the strongly degenerate operator, [Formula: see text] is allowed to be sign-changing, [Formula: see text], [Formula: see text] is a perturbation and the nonlinearity [Formula: see text] is a continuous function does not satisfy the Ambrosetti–Rabinowitz superquadratic condition ((AR) for short). First, via the mountain pass theorem and the Ekeland’s variational principle, existence of two different solutions for [Formula: see text] are obtained when [Formula: see text] satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for [Formula: see text] if [Formula: see text] is odd in [Formula: see text] thanks an extension of Clark’s theorem near the origin. So, our main results considerably improve results appearing in the literature.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

scholarly journals A Flower-Shape Geometry and Nonlinear Problems on Strip-Like Domains

2020 ◽  
Author(s):  
Giuseppe Devillanova ◽  
Giovanni Molica Bisci ◽  
Raffaella Servadei
Keyword(s):  
Symmetric Functions ◽  
Geometrical Structure ◽  
Nonlinear Problems ◽  
Entire Space ◽  
Lebesgue Spaces ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Partially Symmetric

AbstractIn the present paper, we show how to define suitable subgroups of the orthogonal group $${O}(d-m)$$ O ( d - m ) related to the unbounded part of a strip-like domain $$\omega \times {\mathbb {R}}^{d-m}$$ ω × R d - m with $$d\ge m+2$$ d ≥ m + 2 , in order to get “mutually disjoint” nontrivial subspaces of partially symmetric functions of $$H^1_0(\omega \times {\mathbb {R}}^{d-m})$$ H 0 1 ( ω × R d - m ) which are compactly embedded in the associated Lebesgue spaces. As an application of the introduced geometrical structure, we prove (existence and) multiplicity results for semilinear elliptic problems set in a strip-like domain, in the presence of a nonlinearity which either satisfies the classical Ambrosetti–Rabinowitz condition or has a sublinear growth at infinity. The main theorems of this paper may be seen as an extension of existence and multiplicity results, already appeared in the literature, for nonlinear problems set in the entire space $${\mathbb {R}}^d$$ R d , as for instance, the ones due to Bartsch and Willem. The techniques used here are new.

Topological tools for the prescribed scalar curvature problem on S n

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Dina Abuzaid ◽  
Randa Ben Mahmoud ◽  
Hichem Chtioui ◽  
Afef Rigane
Keyword(s):  
Scalar Curvature ◽  
Conformal Metrics ◽  
Standard Sphere ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Hopf Formula ◽  
Curvature Problem ◽  
Formula Type

AbstractIn this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].

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