MULTIPLE SIGN CHANGING SOLUTIONS OF A QUASILINEAR ELLIPTIC EIGENVALUE PROBLEM INVOLVING THE p-LAPLACIAN

We consider the eigenvalue problem [Formula: see text] where Ω⊂ℝN is a bounded smooth domain and Δpu denotes the p-Laplacian, 1<p<+∞; λ>0 is a parameter. The nonlinearity f is required to have an oscillatory behaviour. We prove the existence of multiple positive, multiple negative, and in particular, of multiple sign changing solutions depending on λ.

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Sign-changing solutions for a Schrödinger–Kirchhoff–Poisson system with 4-sublinear growth nonlinearity

Electronic journal of qualitative theory of differential equations ◽

10.14232/ejqtde.2021.1.86 ◽

2021 ◽

pp. 1-21

Author(s):

Shubin Yu ◽

Ziheng Zhang ◽

Rong Yuan

Keyword(s):

Type System ◽

Invariant Sets ◽

Smooth Domain ◽

Positive Parameter ◽

Poisson System ◽

Bounded Smooth Domain ◽

Existence And Multiplicity ◽

Bounded Smooth ◽

Sign Changing Solutions ◽

Poisson Type

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system { − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u + λ ϕ u = Q ( x ) | u | p − 2 u in Ω , − Δ ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , where Ω is a bounded smooth domain of R 3 , a > 0 , b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q ( x ) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of sign-changing solutions to this problem for the case that 2 < p < 4 as λ sufficient small. Furthermore, for λ = 1 and the above assumptions on Q ( x ) , we obtain the same conclusions with 2 < p < 12 5 .

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Multiple Sign-Changing Solutions for Kirchhoff-Type Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2015/985986 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Xingping Li ◽

Xiumei He

Keyword(s):

Mountain Pass ◽

Smooth Domain ◽

Three Solutions ◽

Bounded Smooth Domain ◽

Kirchhoff Type ◽

Bounded Smooth ◽

Sign Changing Solutions

We study the following Kirchhoff-type equations-a+b∫Ω∇u2dxΔu+Vxu=fx,u, inΩ,u=0, in∂Ω, whereΩis a bounded smooth domain ofRN (N=1,2,3),a>0,b≥0,f∈C(Ω¯×R,R), andV∈C(Ω¯,R). Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, iffis odd with respect to its second variable, this problem has infinitely many sign-changing solutions.

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The eigenvalue problem for thep-Laplacian-like equations

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203006744 ◽

2003 ◽

Vol 2003 (9) ◽

pp. 575-586 ◽

Cited By ~ 3

Author(s):

Zu-Chi Chen ◽

Tao Luo

Keyword(s):

Eigenvalue Problem ◽

Smooth Domain ◽

Multiplicity Result ◽

Bounded Smooth Domain ◽

Capillary Phenomena ◽

Bounded Smooth

We consider the eigenvalue problem for the followingp-Laplacian-like equation:−div(a(|Du|p)|Du|p−2Du)=λf(x,u)inΩ,u=0on∂Ω, whereΩ⊂ℝnis a bounded smooth domain. Whenλis small enough, a multiplicity result for eigenfunctions are obtained. Two examples from nonlinear quantized mechanics and capillary phenomena, respectively, are given for applications of the theorems.

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Towers of Nodal Bubbles for the Bahri–Coron Problem in Punctured Domains

International Mathematics Research Notices ◽

10.1093/imrn/rnx289 ◽

2018 ◽

Vol 2019 (19) ◽

pp. 5953-5974

Author(s):

Mónica Clapp ◽

Jorge Faya ◽

Filomena Pacella

Keyword(s):

Blow Up ◽

Critical Sobolev Exponent ◽

Limit Problem ◽

Smooth Domain ◽

Critical Problem ◽

Nodal Solutions ◽

Bounded Smooth Domain ◽

Bounded Smooth ◽

Sobolev Exponent ◽

Sign Changing Solutions

Abstract Let Ω be a bounded smooth domain in $\mathbb {R}^{N}$ which contains a ball of radius R centered at the origin, N ≥ 3. Under suitable symmetry assumptions, for each δ ∈ (0, R), we establish the existence of a sequence (um, δ) of nodal solutions to the critical problem $$\begin{align*}-\Delta u=|u|^{2^{\ast}-2}u\text{ in }\Omega_{\delta}:=\{x\in\Omega :\left\vert x\right\vert>\delta\},\quad u=0\text{ on }\partial \Omega_{\delta},\nonumber\end{align*}$$ where $2^{\ast }:=\frac {2N}{N-2}$ is the critical Sobolev exponent. We show that, if Ω is strictly star-shaped then, for each $m\in \mathbb {N},$ the solutions um, δ concentrate and blow up at 0, as $\delta \rightarrow 0,$ and their limit profile is a tower of nodal bubbles, that is, it is a sum of rescaled nonradial sign-changing solutions to the limit problem $$\begin{align*}-\Delta u=|u|^{2^{\ast}-2}u, \quad u\in D^{1,2}(\mathbb{R}^{N}),\nonumber\end{align*}$$ centered at the origin.

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Further study of a fourth-order elliptic equation with negative exponent

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210509001061 ◽

2011 ◽

Vol 141 (3) ◽

pp. 537-549 ◽

Cited By ~ 2

Author(s):

Zongming Guo ◽

Zhongyuan Liu

Keyword(s):

Fourth Order ◽

Blow Up ◽

Smooth Domain ◽

Regular Solutions ◽

Order Problem ◽

Bounded Smooth Domain ◽

Main Technique ◽

Fourth Order Elliptic Equation ◽

Fourth Order Problem ◽

Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

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A semilinear problem with a gradient term in the nonlinearity

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021110 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Ignacio Guerra

Keyword(s):

Unit Ball ◽

Radial Solution ◽

The Other ◽

Smooth Domain ◽

Gradient Term ◽

Radial Solutions ◽

Bounded Smooth Domain ◽

Other Hand ◽

Bounded Smooth ◽

Semilinear Problem

We consider the following semilinear problem with a gradient term in the nonlinearity<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} -\Delta u = \lambda \frac{(1+|\nabla u|^q)}{(1-u)^p}\quad\text{in}\quad\Omega,\quad u>0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda,p,q>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> be a bounded, smooth domain in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>. We prove that when <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a unit ball and <inline-formula><tex-math id="M5">\begin{document}$ p = 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ q\in (0,q^*(N)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ q^*(N)\in (1,2) $\end{document}</tex-math></inline-formula>, we have infinitely many radial solutions for <inline-formula><tex-math id="M8">\begin{document}$ 2\leq N<2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda = \tilde \lambda $\end{document}</tex-math></inline-formula>. On the other hand, for <inline-formula><tex-math id="M10">\begin{document}$ N>2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> there exists a unique radial solution for <inline-formula><tex-math id="M11">\begin{document}$ 0<\lambda<\tilde \lambda $\end{document}</tex-math></inline-formula>.

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Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2020.52 ◽

2020 ◽

pp. 1-19

Author(s):

Zhijun Zhang

Keyword(s):

Asymptotic Behaviour ◽

Unique Solution ◽

Blow Up ◽

Critical Values ◽

Smooth Domain ◽

Dirichlet Problems ◽

Bounded Smooth Domain ◽

Bounded Smooth

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝ n , g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$ , b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

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A strange bounded smooth domain of holomorphy

Bulletin of the American Mathematical Society ◽

10.1090/s0002-9904-1976-13964-x ◽

1976 ◽

Vol 82 (1) ◽

pp. 74-77 ◽

Cited By ~ 9

Author(s):

Klas Diederich ◽

John Erik Fornaess

Keyword(s):

Smooth Domain ◽

Bounded Smooth Domain ◽

Domain Of Holomorphy ◽

Bounded Smooth

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HARDY-SOBOLEV INEQUALITY IN H1(Ω) AND ITS APPLICATIONS

Communications in Contemporary Mathematics ◽

10.1142/s0219199702000713 ◽

2002 ◽

Vol 04 (03) ◽

pp. 409-434 ◽

Cited By ~ 19

Author(s):

ADIMURTHI

Keyword(s):

Boundary Condition ◽

Sobolev Inequality ◽

Neumann Boundary Condition ◽

Remainder Term ◽

Blow Up ◽

Neumann Boundary ◽

Smooth Domain ◽

Bounded Smooth Domain ◽

Eigen Value ◽

Bounded Smooth

In this article, we have determined the remainder term for Hardy–Sobolev inequality in H1(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

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