scholarly journals Solutions to indefinite weakly coupled cooperative elliptic systems

2021 ◽  
pp. 1-16
Author(s):  
Mónica Clapp ◽  
Andrzej Szulkin
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Ground State ◽  
Elliptic Systems ◽  
Nontrivial Solutions ◽  
Dirichlet Eigenvalues ◽  
Weakly Coupled ◽  
Alpha Beta ◽  
Beta 2

We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 + \lambda\beta|u_1|^\alpha|u_2|^{\beta-2}u_2, \\ u_1,u_2\in D^{1,2}_0(\Omega), \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb R^N$, $N\geq 3$, $\kappa_1,\kappa_2\in\mathbb R$, $\mu_1,\mu_2,\lambda> 0$, $\alpha,\beta> 1$, and $\alpha + \beta = p\le 2^*:={2N}/({N-2})$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $\lambda$ sufficiently large. If $p=2^*$ and $\kappa_1,\kappa_2> 0$ we establish the existence of a ground state for $\lambda$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $\kappa_1$ nor $\kappa_2$ are Dirichlet eigenvalues of $-\Delta$ in $\Omega$.

scholarly journals Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation

2021 ◽  
Vol 19 (1) ◽  
pp. 297-305
Author(s):  
Yuting Zhu ◽  
Chunfang Chen ◽  
Jianhua Chen ◽  
Chenggui Yuan
Keyword(s):  
Bounded Domain ◽  
Ground State ◽  
Multiple Solutions ◽  
Nontrivial Solutions ◽  
Ground State Solutions ◽  
Petviashvili Equation

Abstract In this paper, we study the following generalized Kadomtsev-Petviashvili equation u t + u x x x + ( h ( u ) ) x = D x − 1 Δ y u , {u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u, where ( t , x , y ) ∈ R + × R × R N − 1 \left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1} , N ≥ 2 N\ge 2 , D x − 1 f ( x , y ) = ∫ − ∞ x f ( s , y ) d s {D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s , f t = ∂ f ∂ t {f}_{t}=\frac{\partial f}{\partial t} , f x = ∂ f ∂ x {f}_{x}=\frac{\partial f}{\partial x} and Δ y = ∑ i = 1 N − 1 ∂ 2 ∂ y i 2 {\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}} . We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in R N {{\mathbb{R}}}^{N} .

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

2021 ◽  
pp. 1-28
Author(s):  
B. B. V. Maia ◽  
O. H. Miyagaki
Keyword(s):  
Riesz Potential ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Existence And Nonexistence ◽  
Exponential Critical Growth ◽  
Beta 2 ◽  
Image Position

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem \[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \] when $(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$ (if $N\geq 3$ ) and $(N+\alpha )/p + (N+\beta )/q \geq 2N$ (if $N=2$ ), where $I_{\alpha }$ and $I_{\beta }$ denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem \[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \] where $\alpha ,\,\beta \in (0,\,2)$ and $f,\,g$ have exponential critical growth in the Trudinger–Moser sense.

Ground state solutions of Hamiltonian elliptic systems in dimension two

2019 ◽  
Vol 150 (4) ◽  
pp. 1737-1768 ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
João Marcos do Ó ◽  
Jianjun Zhang
Keyword(s):  
Elliptic System ◽  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Existence Results ◽  
Ground State Solutions ◽  
Variational Framework ◽  
Hamiltonian Elliptic System ◽  
Schrödinger Systems ◽  
Image Position

AbstractThe aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).

scholarly journals Infinitely Many Nontrivial Solutions of Resonant Cooperative Elliptic Systems with Superlinear Terms

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Guanwei Chen ◽  
Shiwang Ma
Keyword(s):  
Ground State ◽  
Elliptic Systems ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Ground State Solutions

We study a class of resonant cooperative elliptic systems and replace the Ambrosetti-Rabinowitz superlinear condition with general superlinear conditions. We obtain ground state solutions and infinitely many nontrivial solutions of this system by a generalized Nehari manifold method developed recently by Szulkin and Weth.

scholarly journals Regularity of the extremal solutions associated with elliptic systems

2018 ◽  
Vol 149 (04) ◽  
pp. 1037-1046
Author(s):  
A. Aghajani ◽  
C. Cowan
Keyword(s):  
Convex Function ◽  
Elliptic System ◽  
Bounded Domain ◽  
Elliptic Systems ◽  
Extremal Solution ◽  
Second Order ◽  
Extremal Solutions ◽  
Smooth Bounded Domain ◽  
Order Polynomial ◽  
Image Position

AbstractWe examine the elliptic system given by$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill &amp; {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$where λ, γ are positive parameters, Ω is a smooth bounded domain in ℝNandfis aC2positive, nondecreasing and convex function in [0, ∞) such thatf(t)/t→ ∞ ast→ ∞. Assuming$$0 < \tau _-: = \mathop {\lim \inf }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les \tau _ + : = \mathop {\lim \sup }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les 2,$$we show that the extremal solution (u*,v*) associated with the above system is smooth provided thatN&lt; (2α*(2 − τ+) + 2τ+)/(τ+)max{1, τ+}, where α*&gt; 1 denotes the largest root of the second-order polynomial$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$As a consequence,u*,v* ∈L∞(Ω) forN&lt; 5. Moreover, if τ−= τ+, thenu*,v* ∈L∞(Ω) forN&lt; 10.

Solutions for Semilinear Elliptic Systems with Critical Sobolev Exponent and Hardy Potential

2010 ◽  
Vol 62 (1) ◽  
pp. 19-33
Author(s):  
Mohammed Bouchekif ◽  
Yasmina Nas
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Bounded Domain ◽  
Elliptic Systems ◽  
Critical Sobolev Exponent ◽  
Existence Results ◽  
Hardy Potential ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Sobolev Exponent

AbstractIn this paper we consider an elliptic system with an inverse square potential and critical Sobolev exponent in a bounded domain of ℝN. By variational methods we study the existence results.

scholarly journals On the Ground State to Hamiltonian Elliptic System with Choquard’s Nonlinear Term

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Wenbo Wang ◽  
Jianwen Zhou ◽  
Yongkun Li
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Ground State ◽  
Nonlinear Term ◽  
Smooth Boundary ◽  
Ground State Solution ◽  
State Solution ◽  
Hamiltonian Elliptic System

In the present paper, we consider the following Hamiltonian elliptic system with Choquard’s nonlinear term −Δu+Vxu=∫ΩGvy/x−yβdygv in Ω,−Δv+Vxv=∫ΩFuy/x−yαdyfu in Ω,u=0,v=0 on ∂Ω,where Ω⊂ℝN is a bounded domain with a smooth boundary, 0<α<N, 0<β<N, and F is the primitive of f, similarly for G. By establishing a strongly indefinite variational setting, we prove that the above problem has a ground state solution.

scholarly journals Solutions for a Class of Nonperiodic Superquadratic Hamiltonian Elliptic Systems Involving Gradient Terms

2017 ◽  
Vol 2017 ◽  
pp. 1-15
Author(s):  
Shengzhong Duan ◽  
Xian Wu
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Existence Result ◽  
Elliptic Systems ◽  
Nontrivial Solutions ◽  
Hamiltonian Elliptic System ◽  
Indefinite Problems

In the present paper, we consider the following Hamiltonian elliptic system HES: -Δu+bx·∇u+Vxu=Hvx,u,v,  x∈RN, -Δv-bx·∇v+Vxv=Hux,u,v,  x∈RN. A new existence result of nontrivial solutions is obtained for the system HES via variational methods for strongly indefinite problems, which generalizes some known results in the literatures.

Homogenization of the conormal derivative problem for elliptic systems in Reifenberg domains

2017 ◽  
Vol 20 (02) ◽  
pp. 1650062 ◽  
Author(s):  
Sun-Sig Byun ◽  
Yunsoo Jang
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Elliptic Systems ◽  
Discontinuous Coefficients ◽  
Bounded Mean Oscillation ◽  
Derivative Problem ◽  
Mean Oscillation ◽  
Reifenberg Flat

We study homogenization of the conormal derivative problem for an elliptic system with discontinuous coefficients in a bounded domain. A uniform global [Formula: see text] estimate for [Formula: see text] is obtained under optimal assumptions that the coefficients have a small bounded mean oscillation (BMO) seminorm and the domain is a [Formula: see text]-Reifenberg flat domain whose boundary might be fractal.

Hamiltonian elliptic systems in dimension two with potentials which can vanish at infinity

2018 ◽  
Vol 20 (08) ◽  
pp. 1750053
Author(s):  
Sérgio H. Monari Soares ◽  
Yony R. Santaria Leuyacc
Keyword(s):  
Elliptic System ◽  
Exponential Growth ◽  
Elliptic Systems ◽  
Positive Function ◽  
Linking Theorem ◽  
Nontrivial Solutions ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Hamiltonian Elliptic System

We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system [Formula: see text] where [Formula: see text] is a positive function which can vanish at infinity and be unbounded from above and [Formula: see text] and [Formula: see text] have exponential growth range. The proof involves a truncation argument combined with the linking theorem and a finite-dimensional approximation.

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