scholarly journals Existence of least energy nodal solution for Kirchhoff-type system with Hartree-type nonlinearity

2020 ◽  
Vol 5 (5) ◽  
pp. 4494-4511
Author(s):  
Jin-Long Zhang ◽  
◽  
Da-Bin Wang
Keyword(s):  
Type System ◽  
Nodal Solution ◽  
Kirchhoff Type ◽  
Least Energy

scholarly journals Morse index of radial nodal solutions of Hénon type equations in dimension two

2017 ◽  
Vol 19 (03) ◽  
pp. 1650042 ◽  
Author(s):  
Ederson Moreira dos Santos ◽  
Filomena Pacella
Keyword(s):  
Elliptic Equations ◽  
Morse Index ◽  
Radial Solution ◽  
Index Formula ◽  
Connected Components ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Semilinear Elliptic ◽  
Bounded Sets ◽  
Least Energy

We consider non-autonomous semilinear elliptic equations of the type [Formula: see text] where [Formula: see text] is either a ball or an annulus centered at the origin, [Formula: see text] and [Formula: see text] is [Formula: see text] on bounded sets of [Formula: see text]. We address the question of estimating the Morse index [Formula: see text] of a sign changing radial solution [Formula: see text]. We prove that [Formula: see text] for every [Formula: see text] and that [Formula: see text] if [Formula: see text] is even. If [Formula: see text] is superlinear the previous estimates become [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] denotes the number of nodal sets of [Formula: see text], i.e. of connected components of [Formula: see text]. Consequently, every least energy nodal solution [Formula: see text] is not radially symmetric and [Formula: see text] as [Formula: see text] along the sequence of even exponents [Formula: see text].

scholarly journals Nodal Solutions for Sublinear-Type Problems with Dirichlet Boundary Conditions

2020 ◽  
Author(s):  
Denis Bonheure ◽  
Ederson Moreira dos Santos ◽  
Enea Parini ◽  
Hugo Tavares ◽  
Tobias Weth
Keyword(s):  
Boundary Conditions ◽  
Morse Index ◽  
Symmetric Functions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Nodal Solution ◽  
Axially Symmetric ◽  
Nodal Solutions ◽  
Dirichlet Eigenvalue ◽  
Least Energy

Abstract We consider nonlinear 2nd-order elliptic problems of the type $$\begin{align*} & -\Delta u=f(u)\ \textrm{in}\ \Omega, \qquad u=0\ \textrm{on}\ \partial \Omega, \end{align*}$$where $\Omega $ is an open $C^{1,1}$–domain in ${{\mathbb{R}}}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen–Cahn type $f(s)=\lambda (s-|s|^{p-1}s)$ with $p>1$ and $\lambda>\lambda _2(\Omega )$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e., sign changing) solution and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $\Omega $ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen–Cahn type nonlinearities in case $\Omega $ is either a ball or a square. In particular, we give a complete description of the solution set for $\lambda \sim \lambda _2(\Omega )$, computing the Morse index of the solutions.

scholarly journals Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz-Sobolev space

2016 ◽  
Vol 290 (4) ◽  
pp. 583-603 ◽  
Author(s):  
Giovany M. Figueiredo ◽  
Jefferson A. Santos
Keyword(s):  
Sobolev Space ◽  
Nodal Solution ◽  
Nodal Domains ◽  
Least Energy ◽  
Kirchhoff Problem

scholarly journals The Existence of Least Energy Sign-Changing Solution for Kirchhoff-Type Problem with Potential Vanishing at Infinity

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ting Xiao ◽  
Canlin Gan ◽  
Qiongfen Zhang
Keyword(s):  
Variational Method ◽  
Type Equation ◽  
Previous Literature ◽  
Type Problem ◽  
Nodal Domains ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Kirchhoff Type Problem ◽  
Quantitative Deformation Lemma ◽  
Least Energy

In this paper, we study the Kirchhoff-type equation: − a + b ∫ ℝ 3     ∇ u 2 d x Δ u + V x u = Q x f u , in   ℝ 3 , where a , b > 0 , f ∈ C 1 ℝ 3 , ℝ , and V , Q ∈ C 1 ℝ 3 , ℝ + . V x and Q x are vanishing at infinity. With the aid of the quantitative deformation lemma and constraint variational method, we prove the existence of a sign-changing solution u to the above equation. Moreover, we obtain that the sign-changing solution u has exactly two nodal domains. Our results can be seen as an improvement of the previous literature.

scholarly journals On a class of Kirchhoff equations involving an anisotropic operator and potential

2020 ◽  
Author(s):  
Mohammed Massar
Keyword(s):  
Variational Methods ◽  
Kirchhoff Equations ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Fractional Equations ◽  
Quantitative Deformation Lemma ◽  
Least Energy

AbstractIn this work, we are concerned with a class of fractional equations of Kirchhoff type with potential. Using variational methods and a variant of quantitative deformation lemma, we prove the existence of a least energy sign-changing solution. Moreover, the existence of infinitely many solution is established.

The existence of solutions with prescribed L2-norm for Kirchhoff type system

2017 ◽  
Vol 58 (4) ◽  
pp. 041502
Author(s):  
Xiaofei Cao ◽  
Junxiang Xu ◽  
Jun Wang
Keyword(s):  
Existence Of Solutions ◽  
Type System ◽  
Kirchhoff Type ◽  
L2 Norm

Existence and concentration of nontrivial nonnegative ground state solutions to Kirchhoff-type system with Hartree-type nonlinearity

2018 ◽  
Vol 69 (6) ◽  
Author(s):  
Fuyi Li ◽  
Chunjuan Gao ◽  
Zhanping Liang ◽  
Junping Shi
Keyword(s):  
Ground State ◽  
Type System ◽  
Ground State Solutions ◽  
Kirchhoff Type
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