scholarly journals Multiple positive solutions for a p-Laplace Benci–Cerami type problem (1 < p < 2), via Morse theory

2021 ◽  
pp. 2150065
Author(s):  
Giuseppina Vannella
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Quasilinear Equation ◽  
Multiple Positive Solutions ◽  
Type Problem ◽  
Poincaré Polynomial ◽  
Subcritical Growth ◽  
Quasilinear Problem

Let us consider the quasilinear problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] is a parameter and [Formula: see text] is a continuous function with [Formula: see text], having a subcritical growth. We prove that there exists [Formula: see text] such that, for every [Formula: see text], [Formula: see text] has at least [Formula: see text] solutions, possibly counted with their multiplicities, where [Formula: see text] is the Poincaré polynomial of [Formula: see text]. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on [Formula: see text], approximating [Formula: see text].

scholarly journals Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

2022 ◽  
Vol 40 ◽  
pp. 1-10
Author(s):  
Duong Trong Luyen ◽  
Le Thi Hong Hanh
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Bounded Domain ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Nontrivial Solutions ◽  
Subcritical Growth ◽  
Subelliptic Operator

In this paper, we study the existence of multiple solutions for the boundary value problem\begin{equation}\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad u=0 \quad \mbox{ on } \partial \Omega, \notag\end{equation}where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$ and $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N), $$the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.

NONTRIVIAL SOLUTIONS FOR N-LAPLACIAN EQUATIONS WITH SUB-EXPONENTIAL GROWTH

2013 ◽  
Vol 11 (03) ◽  
pp. 1350005 ◽  
Author(s):  
ZHONG TAN ◽  
FEI FANG
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Exponential Growth ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Orlicz Spaces ◽  
Nontrivial Solutions ◽  
Laplacian Equations

Let Ω be a bounded domain in RNwith smooth boundary ∂Ω. In this paper, the following Dirichlet problem for N-Laplacian equations (N > 1) are considered: [Formula: see text] We assume that the nonlinearity f(x, t) is sub-exponential growth. In fact, we will prove the mapping f(x, ⋅): LA(Ω) ↦ LÃ(Ω) is continuous, where LA(Ω) and LÃ(Ω) are Orlicz spaces. Applying this result, the compactness conditions would be obtained. Hence, we may use Morse theory to obtain existence of nontrivial solutions for problem (N).

scholarly journals A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces

2009 ◽  
Vol 104 (1) ◽  
pp. 132 ◽  
Author(s):  
Mihai Mihailescu ◽  
Vicentiu Radulescu
Keyword(s):  
Continuous Function ◽  
Eigenvalue Problem ◽  
Continuous Spectrum ◽  
Sobolev Spaces ◽  
Differential Operators ◽  
Smooth Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Sufficient Conditions ◽  
Open Set ◽  
Quasilinear Problem

We study the nonlinear eigenvalue problem $-(\mathrm{div} (a(|\nabla u|)\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in ${\mathsf R}^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.

EIGENVALUE PROBLEMS ASSOCIATED WITH NONHOMOGENEOUS DIFFERENTIAL OPERATORS, IN ORLICZ–SOBOLEV SPACES

2008 ◽  
Vol 06 (01) ◽  
pp. 83-98 ◽  
Author(s):  
MIHAI MIHĂILESCU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Continuous Function ◽  
Boundary Value Problem ◽  
Real Number ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Positive Real Number ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Smooth Boundary ◽  
Positive Real

We study the boundary value problem - div ((a1(|∇ u|) + a2(|∇ u|))∇ u) = λ|u|q(x)-2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN (N ≥ 3) with smooth boundary, λ is a positive real number, q is a continuous function and a1, a2 are two mappings such that a1(|t|)t, a2(|t|)t are increasing homeomorphisms from ℝ to ℝ. We establish the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any λ ∈ [λ1, ∞) is an eigenvalue, while any λ ∈ (0, λ0) is not an eigenvalue of the above problem.

scholarly journals Bifurcation analysis for a modified quasilinear equation with negative exponent

2021 ◽  
Vol 11 (1) ◽  
pp. 684-701
Author(s):  
Siyu Chen ◽  
Carlos Alberto Santos ◽  
Minbo Yang ◽  
Jiazheng Zhou
Keyword(s):  
Bounded Domain ◽  
Bifurcation Analysis ◽  
Bifurcation Theory ◽  
Quasilinear Equation ◽  
Continuous Functions ◽  
Solution Curve ◽  
Bifurcation Parameter ◽  
Smooth Bounded Domain ◽  
Quasilinear Problem ◽  
Bounded Continuous Functions

Abstract In this paper, we consider the following modified quasilinear problem: − Δ u − κ u Δ u 2 = λ a ( x ) u − α + b ( x ) u β i n Ω , u > 0 i n Ω , u = 0 o n ∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a, b are two bounded continuous functions, α > 0, 1 < β ≤ 22* − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

scholarly journals Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

2018 ◽  
Vol 9 (1) ◽  
pp. 108-123 ◽  
Author(s):  
Claudianor O. Alves ◽  
Grey Ercole ◽  
M. Daniel Huamán Bolaños
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Subcritical Growth ◽  
Ground State Solutions ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

Abstract We prove the existence of at least one ground state solution for the semilinear elliptic problem \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=u^{p(x)-1},\quad u% >0,\quad\text{in}\ G\subseteq\mathbb{R}^{N},\ N\geq 3,\\ \displaystyle u&\displaystyle\in D_{0}^{1,2}(G),\end{aligned}\right. where G is either {\mathbb{R}^{N}} or a bounded domain, and {p\colon G\to\mathbb{R}} is a continuous function assuming critical and subcritical values.

An ecological model with the p-Laplacian and diffusion

2015 ◽  
Vol 09 (01) ◽  
pp. 1650008
Author(s):  
S. H. Rasouli
Keyword(s):  
Continuous Function ◽  
Positive Solution ◽  
Bounded Domain ◽  
Population Model ◽  
Population Biology ◽  
Ecological Model ◽  
Smooth Boundary ◽  
Laplacian Operator ◽  
Existence Of Positive Solutions ◽  
And Diffusion

We study the existence of positive solutions of a population model with diffusion of the form [Formula: see text] where Δp denotes the p-Laplacian operator defined by Δpz = div (∣∇z∣p-2∇z), p > 1, Ω is a bounded domain of RN with smooth boundary, α ∈ (0, 1), a and c are positive constants. Here f : [0, ∞) → R is a continuous function. This model arises in the studies of population biology of one species with u representing the concentration of the species. We discuss the existence of positive solution when f satisfies certain additional conditions. We use the method of sub- and super-solutions to establish our results.

scholarly journals Multiplicity of Positive Solutions for a Quasilinear Schrödinger Equation with an Almost Critical Nonlinearity

2020 ◽  
Vol 20 (4) ◽  
pp. 933-963
Author(s):  
Giovany M. Figueiredo ◽  
Uberlandio B. Severo ◽  
Gaetano Siciliano
Keyword(s):  
Positive Solutions ◽  
Existence Result ◽  
Topological Invariants ◽  
Multiple Positive Solutions ◽  
Quasilinear Schrödinger Equation ◽  
Poincaré Polynomial ◽  
Semilinear Equations ◽  
Classical Procedure ◽  
Multiplicity Of Positive Solutions ◽  
Quasilinear Problem

AbstractIn this paper we prove an existence result of multiple positive solutions for the following quasilinear problem:\left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&\displaystyle=|u|% ^{p-2}u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right.where Ω is a smooth and bounded domain in {\mathbb{R}^{N},N\geq 3}. More specifically we prove that, for p near the critical exponent {22^{*}=4N/(N-2)}, the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.

scholarly journals Existence of multiple positive solutions for singular p-q-Laplacian problems with critical nonlinearities

2021 ◽  
Author(s):  
Wang Jiayu ◽  
Wei Han
Keyword(s):  
Weak Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Positive Constant ◽  
Smooth Boundary ◽  
Multiple Positive Solutions ◽  
Critical Nonlinearity ◽  
Critical Nonlinearities ◽  
Existence And Multiplicity ◽  
Alpha Beta

In this article, we consider the following p-q-Laplacian system with singular and critical nonlinearity \begin{equation*} \left \{ \begin{array}{lllll} -\Delta_{p}u-\Delta_{q}u=\frac{h_{1}(x)}{u^{r}}+\lambda\frac{\alpha}{\alpha+\beta}u^{\alpha-1}v^{\beta} \ \ in\ \Omega ,\\ -\Delta_{p}v-\Delta_{q}v=\frac{h_{2}(x)}{v^{r}}+\lambda\frac{\beta}{\alpha+\beta}u^{\alpha}v^{\beta-1} \ \ in\ \Omega, \\ u,v>0 \ \ \ \ \ \ in \ \Omega, \ \ \ \ \ u=v=0 \ \ \ \ \ \ \ on \ \partial\Omega, \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb {R}^{n}$ with smooth boundary $\partial\Omega$. $11,\lambda\in(0,\Lambda_{*})$ is parameter with $\Lambda _{*}$ is a positive constant and $h_{1}(x),h_{2}(x)\in L^{\infty},h_{1}(x),h_{2}(x)>0$. We show the existence and multiplicity of weak solution of equation above for suitable range of $\lambda$.

scholarly journals Sign-changing solutions for Schrödinger system with critical growth

2021 ◽  
Vol 30 (1) ◽  
pp. 242-256
Author(s):  
Changmu Chu ◽  
◽  
Jiaquan Liu ◽  
Zhi-Qiang Wang ◽  
◽  
...  
Keyword(s):  
Bounded Domain ◽  
Additional Condition ◽  
Smooth Boundary ◽  
Invariant Sets ◽  
Nonlinear Coupling ◽  
Coupling Matrix ◽  
Subcritical Growth ◽  
Coupling Terms ◽  
Sign Changing Solutions ◽  
Schrödinger System

<abstract><p>We consider the following Schrödinger system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{aligned} &amp;-\Delta u_j = \sum\limits_{i = 1}^k \beta_{ij}|u_i|^3|u_j|u_j+\lambda_j|u_j|^{q-2}u_j, \ \ \ \text{in}\, \, \Omega, \\ &amp;u_j = 0\quad\text{on}\, \, \partial\Omega, \, \, j = 1, \cdots, k \end{aligned} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \Omega\subset\mathbb{R}^3 $ is a bounded domain with smooth boundary. Assume $ 5 &lt; q &lt; 6, \, \lambda_j &gt; 0, \, \beta_{jj} &gt; 0, \, j = 1, \cdots, k $, $ \beta_{ij} = \beta_{ji}, \, i\neq j, i, j = 1, \cdots, k $. Note that the nonlinear coupling terms are of critical Sobolev growth in dimension $ 3 $. We prove that under an additional condition on the coupling matrix the problem has infinitely many sign-changing solutions. The result is obtained by combining the method of invariant sets of descending flow with the approach of using approximation of systems of subcritical growth.</p></abstract>

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