Equations of p-Laplacian Type in Unbounded Domains

2002 ◽  
Vol 2 (3) ◽  
Author(s):  
Pablo L. De Nápoli ◽  
M. Cristina Mariani
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Spaces ◽  
Unbounded Domains ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Solutions To Equations

AbstractThis work is devoted to study the existence of solutions to equations of the p Laplacian type in unbounded domains. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. We apply the mountain pass theorem in weighted Sobolev spaces.

scholarly journals Existence and multiplicity of nontrivial solutions for nonlinear Schrödinger equations with unbounded potentials

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2465-2481 ◽  
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Bitao Cheng ◽  
Huxiao Luo
Keyword(s):  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Spaces ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Compact Embedding ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Unbounded Potential ◽  
Superquadratic Growth

We investigate the existence of nontrivial solutions and multiple solutions for the following class of elliptic equations (-?u + V(x)u = K(x)f(u), x ? RN, u ? D1,2(RN), where N ? 3, V(x) and K(x) are both unbounded potential functions and f is a function with a superquadratic growth. Firstly, we prove the existence of infinitely many solutions with compact embedding and by means of symmetric mountain pass theorem. Moreover, we prove the existence of nontrivial solutions without compact embedding in weighted Sobolev spaces and by means of mountain pass theorem. Our results extend and generalize some existing results.

Existence results for a Kirchhoff type equation in Orlicz–Sobolev spaces

2017 ◽  
Vol 8 (3) ◽  
Author(s):  
EL Miloud Hssini ◽  
Najib Tsouli ◽  
Mustapha Haddaoui
Keyword(s):  
Variational Principle ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Type Equation ◽  
Existence Results ◽  
Mountain Pass ◽  
Nonlocal Problems ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation

AbstractIn this paper, based on the mountain pass theorem and Ekeland’s variational principle, we show the existence of solutions for a class of non-homogeneous and nonlocal problems in Orlicz–Sobolev spaces.

scholarly journals Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign

2015 ◽  
Vol 4 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Ziheng Zhang ◽  
Rong Yuan
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Fractional Hamiltonian Systems ◽  
Symmetric Mountain Pass Theorem

AbstractIn this paper we are concerned with the existence of infinitely-many solutions for fractional Hamiltonian systems of the form ${\,}_tD^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}u(t))+L(t)u(t)=\nabla W(t,u(t))$, where ${\alpha \in (\frac{1}{2},1)}$, ${t\in \mathbb {R}}$, ${u\in \mathbb {R}^n}$, ${L\in C(\mathbb {R},\mathbb {R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in \mathbb {R}}$, ${W\in C^1(\mathbb {R}\times \mathbb {R}^n,\mathbb {R})}$ and ${\nabla W(t,u)}$ is the gradient of ${W(t,u)}$ at u. The novelty of this paper is that, assuming L(t) is bounded in the sense that there are constants ${0<\tau _1<\tau _2< \infty }$ such that ${\tau _1 |u|^2\le (L(t)u,u)\le \tau _2 |u|^2}$ for all ${(t,u)\in \mathbb {R}\times \mathbb {R}^n}$ and ${W(t,u)}$ is of the form ${({a(t)}/({p+1}))|u|^{p+1}}$ such that ${a\in L^{\infty }(\mathbb {R},\mathbb {R})}$ can change its sign and ${0<p<1}$ is a constant, we show that the above fractional Hamiltonian systems possess infinitely-many solutions. The proof is based on the symmetric mountain pass theorem. Recent results in the literature are generalized and significantly improved.

scholarly journals Existence of solutions for a boundary problem involving p(x)-biharmonic operator

2012 ◽  
Vol 31 (1) ◽  
pp. 179 ◽  
Author(s):  
Abdel Rachid El Amrouss ◽  
Anass Ourraoui
Keyword(s):  
Boundary Problem ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Biharmonic Operator ◽  
Technical Approach ◽  
Three Solutions

In this paper, we establish the existence of at least three solutions to a boundary problem involving the p(x)-biharmonic operator. Our technical approach is based on theorem obtained by B. Ricceri's variational principale and local mountain pass theorem without (Palais.Smale) condition.

scholarly journals SOLUTIONS OF THE DIVERGENCE AND ANALYSIS OF THE STOKES EQUATIONS IN PLANAR HÖLDER-α DOMAINS

2010 ◽  
Vol 20 (01) ◽  
pp. 95-120 ◽  
Author(s):  
RICARDO G. DURÁN ◽  
FERNANDO LÓPEZ GARCÍA
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Basic Result ◽  
Weighted Sobolev Spaces ◽  
Mean Value ◽  
Well Posedness ◽  
Existence Of A Solution ◽  
Simply Connected ◽  
Distance To The Boundary

If Ω ⊂ ℝn is a bounded domain, the existence of solutions [Formula: see text] of div u = f for f ∈ L2(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular, it allows to show the existence of a solution [Formula: see text], where u is the velocity and p the pressure. It is known that the above-mentioned result holds when Ω is a Lipschitz domain and that it is not valid for arbitrary Hölder-α domains. In this paper we prove that if Ω is a planar simply connected Hölder-α domain, there exist solutions of div u = f in appropriate weighted Sobolev spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. For some particular domains with an external cusp, we apply our results to show the well-posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution [Formula: see text] for some r < 2 depending on the power of the cusp.

scholarly journals Multiple solutions for some strongly degenerate second order elliptic equations

2021 ◽  
pp. 1-12
Author(s):  
João R. Santos ◽  
Gaetano Siciliano
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Degenerate Operator ◽  
Second Order Elliptic Equations ◽  
Strongly Degenerate

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form L ( u ) = − div ( a ( x ) ∇ u ) and a suitable nonlinearity f. The function a vanishes on smooth 1-codimensional submanifolds of Ω where it is not allowed to be C 2 . By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where a vanishes.

scholarly journals Local existence of solutions for the modified Korteweg-de Vries (mKdV) equation in weighted Sobolev spaces

2021 ◽  
Vol 2 (1) ◽  
pp. 458-466
Author(s):  
Milton M. Cortez Gutiérrez ◽  
Hernan O. Cortez Gutiérrez ◽  
Girady I. Cortez Fuentes Rivera ◽  
Liv J. Cortez Fuentes Rivera ◽  
Deolinda E. Fuentes Rivera Vallejo
Keyword(s):  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Weighted Sobolev Spaces ◽  
Mkdv Equation ◽  
De Vries ◽  
Local Existence Of Solutions
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