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.

Multiple Solutions of Sublinear Quasilinear Schrödinger Equations with Small Perturbations

2018 ◽  
Vol 62 (2) ◽  
pp. 471-488
Author(s):  
Liang Zhang ◽  
X. H. Tang ◽  
Yi Chen
Keyword(s):  
Critical Point Theory ◽  
Multiple Solutions ◽  
Point Theory ◽  
Energy Functional ◽  
Negative Energy ◽  
Quasilinear Schrödinger Equation ◽  
Small Perturbations ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Image Position

AbstractIn this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation $$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$ where Ω is a bounded smooth domain in ℝN (N ≥ 1), α ≥ 2 and θ is a parameter. Under the assumption that g(x, u) is sublinear near the origin with respect to u, we study the effect of the perturbation term h(x, u), which may break the symmetry of the associated energy functional. With the aid of critical point theory and the truncation method, we show that this system possesses multiple small negative energy solutions.

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.

On inhomogeneous biharmonic equations involving critical exponents

1999 ◽  
Vol 129 (5) ◽  
pp. 925-946 ◽  
Author(s):  
Yinbin Deng ◽  
Gengsheng Wang
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Mountain Pass Lemma ◽  
Biharmonic Equations ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Image Position

In this paper, we consider the existence of multiple solutions of biharmonic equations boundary value problemwhere Ω is a bounded smooth domain in ℝN, N ≥ 5; λ ∈ ℝ1 is a given constant; p = 2N/(N − 4) is the critical Sobolev exponent for the embedding ; Δ2 = ΔΔ denotes iterated N-dimensional Laplacian; f(x) is a given function. Some results on the existence and non-existence of multiple solutions for the above problem have been obtained by Ekeland's variational principle and the mountain-pass lemma under some assumptions on f(x) and N.

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.

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

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