scholarly journals A Class of Equations with Three Solutions

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 478 ◽  
Author(s):  
Biagio Ricceri
Keyword(s):  
Bounded Domain ◽  
Weak Solutions ◽  
Convex Set ◽  
First Eigenvalue ◽  
Global Minima ◽  
Three Solutions ◽  
Smooth Bounded Domain ◽  
The First Eigenvalue

Here is one of the results obtained in this paper: Let Ω ⊂ R n be a smooth bounded domain, let q > 1 , with q < n + 2 n - 2 if n ≥ 3 and let λ 1 be the first eigenvalue of the problem - Δ u = λ u in Ω , u = 0 on ∂ Ω . Then, for every λ > λ 1 and for every convex set S ⊆ H 0 1 ( Ω ) dense in H 0 1 ( Ω ) , there exists α ∈ S such that the problem - Δ u = λ ( u + - ( u + ) q ) + α ( x ) in Ω , u = 0 on ∂ Ω , has at least three weak solutions, two of which are global minima in H 0 1 ( Ω ) of the functional u → 1 2 ∫ Ω | ∇ u ( x ) | 2 d x - λ ∫ Ω 1 2 | u + ( x ) | 2 - 1 q + 1 | u + ( x ) | q + 1 d x - ∫ Ω α ( x ) u ( x ) d x where u + = max { u , 0 } .

scholarly journals A class of functionals possessing multiple global minima

2021 ◽  
Vol 66 (1) ◽  
pp. 75-84
Author(s):  
Biagio Ricceri
Keyword(s):  
Bounded Domain ◽  
Weak Solutions ◽  
Convex Set ◽  
Multiplicity Result ◽  
Global Minima ◽  
Smooth Bounded Domain ◽  
Gradient Systems ◽  
Alpha Beta

"We get a new multiplicity result for gradient systems. Here is a very particular corollary: Let $\Omega\subset {\bf R}^n$ ($n\geq 2$) be a smooth bounded domain and let $\Phi:{\bf R}^2\to {\bf R}$ be a $C^1$ function, with $\Phi(0,0)=0$, such that $$\sup_{(u,v)\in {\bf R}^2}\frac{|\Phi_u(u,v)|+|\Phi_v(u,v)|}{1+|u|^p+|v|^p}<+\infty$$ where $p>0$, with $p=\frac{2}{n-2}$ when $n>2$.\\ Then, for every convex set $S\subseteq L^{\infty}(\Omega)\times L^{\infty}(\Omega)$ dense in $L^2(\Omega)\times L^2(\Omega)$, there exists $(\alpha,\beta)\in S$ such that the problem $$-\Delta u=(\alpha(x)\cos(\Phi(u,v))-\beta(x)\sin(\Phi(u,v)))\Phi_u(u,v)\hskip 5pt \hbox {\rm in}\hskip 5pt \Omega$$ $$-\Delta v= (\alpha(x)\cos(\Phi(u,v))-\beta(x)\sin(\Phi(u,v)))\Phi_v(u,v)\hskip 5pt \hbox {\rm in}\hskip 5pt \Omega$$ $$u=v=0\hskip 5pt \hbox {\rm on}\hskip 5pt \partial\Omega$$ has at least three weak solutions, two of which are global minima in $H^1_0(\Omega)\times H^1_0(\Omega)$ of the functional $$(u,v)\to \frac{1}{2}\left ( \int_{\Omega}|\nabla u(x)|^2dx+\int_{\Omega}|\nabla v(x)|^2dx\right )$$ $$-\int_{\Omega}(\alpha(x)\sin(\Phi(u(x),v(x)))+\beta(x)\cos(\Phi(u(x),v(x))))dx\ .$$"

scholarly journals Nondegeneracy of solutions for a class of cooperative systems on $ \mathbb{R}^n $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Marcello Lucia ◽  
Guido Sweers
Keyword(s):  
Bounded Domain ◽  
Cooperative Systems ◽  
First Eigenvalue ◽  
Exponential Nonlinearity ◽  
The First Eigenvalue ◽  
Chern Simons ◽  
Fully Coupled

<p style='text-indent:20px;'>We consider fully coupled cooperative systems on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> with exponential nonlinearity, are nondegenerate.</p>

scholarly journals Eigenvalues for a combination between local and nonlocal p-Laplacians

2019 ◽  
Vol 22 (5) ◽  
pp. 1414-1436 ◽  
Author(s):  
Leandro M. Del Pezzo ◽  
Raúl Ferreira ◽  
Julio D. Rossi
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Limit Problem ◽  
First Eigenvalue ◽  
Order Zero ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue ◽  
Homogeneous Operator ◽  
Dirichlet Eigenvalue Problem

Abstract In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝN, Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.

scholarly journals Twin Positive Solutions for Schrödinger-Kirchhoff-Type Problem with Singularity and Critical Exponents

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Wei Han ◽  
Yangyang Zhao
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Perturbation Methods ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
The First Eigenvalue ◽  
Kirchhoff Type Problem

We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent -a+b∫Ω∇u2dxΔu+u=Q(x)u5+μxα-2u+f(x)(λ/uγ) in Ω,u=0 on ∂Ω, where a,b>0 are constants, Ω⊂R3 is a smooth bounded domain, 0<α<1, λ>0 is a real parameter, γ∈(0,1) is a constant, and 0<μ<aμ1 (μ1 is the first eigenvalue of -Δu=μxα-2u, under Dirichlet boundary condition). Under appropriate assumptions on Q and f, we obtain two positive solutions via the variational and perturbation methods.

scholarly journals Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tuhina Mukherjee ◽  
Patrizia Pucci ◽  
Mingqi Xiang
Keyword(s):  
Bounded Domain ◽  
Weak Solutions ◽  
Nehari Manifold ◽  
Real Parameter ◽  
Combined Effects ◽  
Smooth Bounded Domain ◽  
Existence Proofs ◽  
The Nehari Manifold ◽  
Kirchhoff Problem

<p style='text-indent:20px;'>In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} M\left(\|u\|^{{n}/{s}}\right)(-\Delta)^s_{n/s}u = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\quad\text{in } \Omega,\\ u&gt;0\qquad\text{in } \Omega,\\ u = 0\qquad\text{in } \mathbb R^n \setminus{ \Omega}, \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu&gt;0 $\end{document}</tex-math></inline-formula> is a real parameter, <inline-formula><tex-math id="M6">\begin{document}$ \beta &lt;{n/(n-s)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q\in (0,1) $\end{document}</tex-math></inline-formula>.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.</p>

scholarly journals The existence of nontrivial solutions for a critically coupled Schr\”{o}dinger system in a bounded domain of $\R^3$

2021 ◽  
Author(s):  
Hongyu Ye ◽  
Lina Zhang
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
The First Eigenvalue ◽  
Nirenberg Problem

In this paper, we consider the following coupled Schr\”{o}dinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg problem $$\left\{% \begin{array}{ll} -\Delta u+\lambda_1 u=\mu_1 u^5+ \beta u^2v^3, & \hbox{$x\in \Omega$}, \\ -\Delta v+\lambda_2 v=\mu_2 v^5+ \beta v^2u^3, & \hbox{$x\in \Omega$}, \\ u=v=0,& \hbox{$x\in \partial\Omega$}, \\ \end{array}% \right.$$ where $\Omega$ is a ball in $\R^3,$ $-\lambda_1(\Omega)<\lambda_1,\lambda_2<-\frac14\lambda_1(\Omega)$, $\mu_1,\mu_2>0$ and $\beta>0$. Here $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition in $\Omega$. We show that the problem has at least one nontrivial solution for all $\beta>0$.

scholarly journals On the existence threshold for positive solutions of p-Laplacian equations with a concave–convex nonlinearity

2015 ◽  
Vol 17 (06) ◽  
pp. 1450044 ◽  
Author(s):  
Fernando Charro ◽  
Enea Parini
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Boundary Value ◽  
First Eigenvalue ◽  
Limit Case ◽  
The First Eigenvalue ◽  
Laplacian Equations

We study the following boundary value problem with a concave–convex nonlinearity: [Formula: see text] Here Ω ⊂ ℝnis a bounded domain and 1 < q < p < r < p*. It is well known that there exists a number Λq, r> 0 such that the problem admits at least two positive solutions for 0 < Λ < Λq, r, at least one positive solution for Λ = Λq, r, and no positive solution for Λ > Λq, r. We show that [Formula: see text] where λ1(p) is the first eigenvalue of the p-Laplacian. It is worth noticing that λ1(p) is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case q = p.

scholarly journals Generalized Picone inequalities and their applications to (p,q)-Laplace equations

2020 ◽  
Vol 18 (1) ◽  
pp. 1030-1044
Author(s):  
Vladimir Bobkov ◽  
Mieko Tanaka
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Positive Solutions ◽  
First Eigenvalue ◽  
Resonance Case ◽  
The First Eigenvalue ◽  
Laplace Equations ◽  
Existence And Nonexistence ◽  
Laplace Type ◽  
Picone Inequality

Abstract We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the (p,q) -Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation -\hspace{-0.25em}{\text{&#x0394;}}_{p}u-{\text{&#x0394;}}_{q}u={f}_{\mu }(x,u,\nabla u) in a bounded domain \text{&#x03A9;}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} under certain assumptions on the nonlinearity and with a special attention to the resonance case {f}_{\mu }(x,u,\nabla u)={\lambda }_{1}(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u , where {\lambda }_{1}(p) is the first eigenvalue of the p-Laplacian.

EIGENVALUE ESTIMATES ON DOMAIN OF THE POLYDISK

2012 ◽  
Vol 23 (01) ◽  
pp. 1250014
Author(s):  
TAO ZHENG ◽  
DAGUANG CHEN ◽  
MIN CAI
Keyword(s):  
Bounded Domain ◽  
First Eigenvalue ◽  
Dirichlet Laplacian ◽  
Clamped Plate ◽  
Eigenvalue Estimates ◽  
The First Eigenvalue ◽  
Universal Inequalities

In this paper, we investigate universal inequalities for eigenvalues of the Dirichlet Laplacian and the clamped plate problem on a bounded domain in an n-dimensional polydisk 𝔻n. Moreover, from the domain monotonicity of the eigenvalue, we can prove that if the first eigenvalue of the Dirichlet Laplacian tends to [Formula: see text] when the domain tends to the polydisk 𝔻n, then all of the eigenvalues tend to [Formula: see text].

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

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