Linked eigenvalue problems for the p-Laplacian

1994 ◽  
Vol 124 (5) ◽  
pp. 1023-1036
Author(s):  
P. A. Binding ◽  
Y. X. Huang
Keyword(s):  
Existence And Uniqueness ◽  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Smooth Domain ◽  
Uniqueness Of Solutions ◽  
Oscillation Theorem ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Image Position

Linked equations of the formare studied on a bounded smooth domain in RN for λ ∈ R2. Existence and uniqueness of solutions are discussed for fi homogeneous of order p – 1 in ui, generalising the ‘Klein Oscillation Theorem’ when p = 2, N = 1. Bifurcation from the principal eigenvalue is also considered for nonhomogeneous perturbations fi of order greater than p – 1.

A short remark regarding Pohozaev-type results on general domains assuming finite Morse index

2017 ◽  
Vol 147 (2) ◽  
pp. 293-297 ◽  
Author(s):  
Craig Cowan
Keyword(s):  
Morse Index ◽  
Negative Integer ◽  
Smooth Domain ◽  
Negative Solution ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Image Position

We are interested in non-negative non-trivial solutions ofwhere 1 < p and Ω is a bounded smooth domain in ℝN with 3 ⩽ N ⩽ 9. We show that given a non-negative integer M there is some large p(M,Ω) such that the only non-negative solution u, of Morse index at most M, is u = 0.

MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH

2017 ◽  
Vol 96 (1) ◽  
pp. 98-109 ◽  
Author(s):  
MARCELO F. FURTADO ◽  
HENRIQUE R. ZANATA
Keyword(s):  
Variational Methods ◽  
Multiple Solutions ◽  
Kirchhoff Equation ◽  
Smooth Domain ◽  
Infinitely Many Solutions ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Image Position

We prove the existence of infinitely many solutions $u\in W_{0}^{1,2}(\unicode[STIX]{x1D6FA})$ for the Kirchhoff equation $$\begin{eqnarray}\displaystyle -\biggl(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{2}\,dx\biggr)\unicode[STIX]{x1D6E5}u=a(x)|u|^{q-1}u+\unicode[STIX]{x1D707}f(x,u)\quad \text{in }\unicode[STIX]{x1D6FA}, & & \displaystyle \nonumber\end{eqnarray}$$ where $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}$ is a bounded smooth domain, $a(x)$ is a (possibly) sign-changing potential, $0<q<1$, $\unicode[STIX]{x1D6FC}>0$, $\unicode[STIX]{x1D6FD}\geq 0$, $\unicode[STIX]{x1D707}>0$ and the function $f$ has arbitrary growth at infinity. In the proof, we apply variational methods together with a truncation argument.

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

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
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 < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 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.

scholarly journals A semilinear problem with a gradient term in the nonlinearity

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

<p style='text-indent:20px;'>We consider the following semilinear problem with a gradient term in the nonlinearity</p><p style='text-indent:20px;'><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&gt;0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda,p,q&gt;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&lt;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&gt;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&lt;\lambda&lt;\tilde \lambda $\end{document}</tex-math></inline-formula>.</p>

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

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.

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

2002 ◽  
Vol 04 (03) ◽  
pp. 409-434 ◽  
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.

scholarly journals Existence and Asymptotic Profile of Nodal Solutions to Supercritical Problems

2017 ◽  
Vol 17 (1) ◽  
Author(s):  
Mónica Clapp ◽  
Filomena Pacella
Keyword(s):  
Smooth Domain ◽  
Nodal Solutions ◽  
Bounded Smooth Domain ◽  
Asymptotic Profile ◽  
Bounded Smooth ◽  
Supercritical Problem

AbstractWe establish the existence of nodal solutions to the supercritical problemin a symmetric bounded smooth domain Ω of

Structure of positive boundary blow-up solutions

2004 ◽  
Vol 134 (5) ◽  
pp. 923-938 ◽  
Author(s):  
Zongming Guo
Keyword(s):  
Boundary Layer ◽  
Blow Up ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Semilinear Problems ◽  
Bounded Smooth ◽  
Positive Boundary

The structure of positive boundary blow-up solutions to semilinear problems of the form −Δu = λf(u) in Ω, u = ∞ on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied for a class of nonlinearities f ∈ C1 ([0, ∞)\{z2}) satisfying f (0) = f(z1) = f (z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1)∪(z2, ∞), f > 0 in (z1, z2). Two positive boundary-layer solutions and infinitely many positive spike-layer solutions are obtained for λ sufficiently large.

scholarly journals Existence of Solutions for a Modified Nonlinear Schrödinger System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yujuan Jiao ◽  
Yanli Wang
Keyword(s):  
Perturbation Method ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Schrödinger System

We are concerned with the following modified nonlinear Schrödinger system:-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu,  x∈Ω,  -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v,  x∈Ω,  u=0,  v=0,  x∈∂Ω, whereα>2,  β>2,  α+β<2·2*,  2*=2N/(N-2)is the critical Sobolev exponent, andΩ⊂ℝN  (N≥3)is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

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