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article

Nonlinear boundary value problems of a class of elliptic equations involving critical variable exponents

Yingying Shan, Yongqiang Fu

Boundary Value Problems > 2019 > 2019 > 1 > 1-21

In this paper, we first obtain the existence of solutions for a class of elliptic equations involving critical variable exponents and nonlinear boundary values by the mountain pass theorem and concentration compactness principle. Then, under suitable assumptions, we obtain a sequence of solutions with positive energies going towards infinity by Fountain Theorem.

article

New characterizations of Musielak-Orlicz-Sobolev spaces via sharp ball averaging functions

Sibei Yang, Dachun Yang, Wen Yuan

Frontiers of Mathematics in China > 2019 > 14 > 1 > 177-201

We establish a new characterization of the Musielak-Orlicz-Sobolev space on ℝⁿ, which includes the classical Orlicz-Sobolev space, the weighted Sobolev space, and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result...

article

Multiplicity of solutions for Schrödinger systems with critical exponent via Ljusternik–Schnirelman theory

Thiziri Chergui, Saadia Tas

Rendiconti del Circolo Matematico di Palermo Series 00002 > 2018 > 67 > 2 > 267-289

In this paper, by exploiting the Nehari manifold method and the Ljusternik–Schnirelman category theory, we establish the existence and multiplicity results of the nonlinear Schrödinger systems of the form $$\begin{aligned} \left\{ \begin{array}{l} -\epsilon ^{p\left( x\right) }div\left( \left| \nabla u\right| ^{p\left( x\right) -2}\nabla u\right) +V\left( z\right) \left| u\right| ^{p\left( x\right)...

article

An integro-differential inequality related to the smallest positive eigenvalue ofp(x)-Laplacian Dirichlet problem

Damian Wiśniewski, Mariusz Bodzioch

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica > 2016 > 15

We consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.

article

A minimization problem with variable growth on Nehari manifold

Xia Zhang

Monatshefte für Mathematik > 2016 > 181 > 2 > 485-500

In this paper, based on the theory of variable exponent space, we study a class of minimizing problem on Nehari manifold via concentration compactness principle. Under suitable assumptions, by showing a relative compactness of minimizing sequences, we prove the existence of minimizers.

article

Density of smooth functions in variable exponent Sobolev spaces

Thanasis Kostopoulos, Nikos Yannakakis

Nonlinear Analysis > 2015 > 127 > Complete > 196-205

We show that if p−≥2, then a sufficient condition for the density of smooth functions with compact support, in the variable exponent Sobolev space W1,p(⋅)(Rn), is that the Riesz potentials of compactly supported functions of Lp(⋅)(Rn), are also elements of Lp(⋅)(Rn). Using this result we then prove that the above density holds if (i) p−≥n or if (ii) 2≤p−<n and p+<np−n−p−. Moreover our result...

article

Existence of three solutions for variable exponent elliptic systems

Mostafa Allaoui

ANNALI DELL'UNIVERSITA' DI FERRARA > 2015 > 61 > 2 > 241-253

We study the solutions of the $$(p(x),q(x))$$ ( p ( x ) , q ( x ) ) -biharmonic system with Navier boundary condition on a bounded domain, and obtain three solutions under appropriate hypotheses. The technical approach is mainly based on a three critical points theorem due to Ricceri.

article

Higher integrability for nonlinear elliptic equations with variable growth and discontinuous coefficients

Xia Zhang, Yan Huo, Yongqiang Fu

Journal of Mathematical Analysis and Applications > 2014 > 418 > 1 > 425-443

In this paper, based on the theory of variable exponent spaces, we study the higher integrability for a class of nonlinear elliptic equations with variable growth and discontinuous coefficients. Under suitable assumptions, we obtain a local gradient estimate in Orlicz space for weak solution.

article

Existence of a nontrival solution for Dirichlet problem involving p(x)-Laplacian

Sylwia Barnaś

Discussiones Mathematicae, Differential Inclusions, Control and Optimization > 2014 > 34 > 1 > 15-39

In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang [6] and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of solution for our problem.

article

Solutions for nonlinear elliptic equations with variable growth and degenerate coercivity

Xia Zhang, Yongqiang Fu

Annali di Matematica Pura ed Applicata ( 01923 -) > 2014 > 193 > 1 > 133-161

In this paper, based on the theory of variable exponent Sobolev space, we study a class of nonlinear elliptic equations with principal part having degenerate coercivity and obtain some existence and regularity results for the solutions.

article

Higher integrability for nonlinear elliptic equations with variable growth

Xia Zhang, Yongqiang Fu

Nonlinear Analysis > 2013 > 93 > Complete > 132-146

In this paper, based on the theory of variable exponent spaces, we study the regularity for a class of nonlinear elliptic equation with p(x)-Laplacian. Under suitable assumptions, we obtain a local gradient estimate in the Orlicz space for weak solution.

article

Variable Exponent Sobolev Spaces for Semilinear Elliptic Systems

Zhong Tan, Fei Fang

Mediterranean Journal of Mathematics > 2013 > 10 > 3 > 1353-1367

In this work, the semilinear elliptic systems with Dirichlet boundary value are considered. We extend the notion of subcritical growth from polynomial growth to variable exponent growth. Under the variable exponent growth, nontrivial solutions are obtained via variable exponent Sobolev spaces and variational methods. In article final, we make a remark to explain that our main result is a extention...

article

Ni-Serrin type equations arising from capillarity phenomena with non-standard growth

Mustafa Avci

Boundary Value Problems > 2013 > 2013 > 1 > 1-13

In the present paper, in view of the variational approach, we discuss a Ni-Serrin type equation involving non-standard growth condition and arising from the capillarity phenomena. Establishing some suitable conditions, we prove the existence and multiplicity of solutions. MSC: 35D05, 35J60, 35J70.

article

Infinitely many homoclinic solutions for nonautonomous p(t)-Laplacian Hamiltonian systems

Peng Chen, X.H. Tang, Ravi P. Agarwal

Computers and Mathematics with Applications > 2012 > 63 > 4 > 751-763

By using the Symmetric Mountain Pass Theorem, we establish some existence criteria which guarantee that the second-order ordinary p(t)-Laplacian systems of the form ddt(∣u̇(t)∣p(t)−2u̇(t))−a(t)∣u(t)∣p(t)−2u(t)+∇W(t,u(t))=0 have infinitely many homoclinic solutions, where t∈R,u∈RN, p∈C(R,R) and p(t)>1, a∈C(R,R), and W∈C1(R×RN,R) are non-periodic in t.

article

Multiplicity of Solutions on a Nonlinear Eigenvalue Problem for p(x)-Laplacian-like Operators

M. Manuela Rodrigues

Mediterranean Journal of Mathematics > 2012 > 9 > 1 > 211-223

The paper study the existence and multiplicity of solutions for the nonlinear eigenvalue problems for p(x)-Laplacian-like operators, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

article

Existence result for hemivariational inequality involving p(x)-Laplacian

S. Barnaś

Opuscula Mathematica > 2012 > Vol. 32, no. 3 > 439-454

In this paper we study the nonlinear elliptic problem with p(x)-Laplacian (hemivariational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang [J. Math. Anal. Appl. 80 (1981), 102-129].

article

Existence Result for Diﬀerential Inclusion with p(x) - Laplacian

S. Barnaś

Schedae Informaticae > 2012 > Vol. 21 > 41--54

In this paper we study the nonlinear elliptic problem with p(x)- Laplacian (hemivariational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang [4].

article

Existence of periodic solutions for a differential inclusion systems involving the p(t)-Laplacian

Ge Bin, Xue Xiaoping, Zhou Qingmei

Acta Mathematica Scientia > 2011 > 31 > 5 > 1786-1802

We study a nonlinear periodic problem driven by the p(t)-Laplacian and having a nonsmooth potential (hemivariational inequalities). Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution...

article

Existence of at least five solutions for a differential inclusion problem involving the p(x)-Laplacian

Bin Ge, Xiaoping Xue, Qingmei Zhou

Nonlinear Analysis: Real World Applications > 2011 > 12 > 4 > 2304-2318

We studied a nonlinear Dirichlet problem driven by the p(x)-Laplacian and having a nonsmooth potential (hemivariational inequalities). Using a variational method combined with suitable truncation techniques based on nonsmooth critical point theory for locally Lipschitz function, we proved the existence of at least five solutions under the suitable conditions.

article

Discontinuous elliptic problems involving the p(x)‐Laplacian

Gabriele Bonanno, Antonia Chinnì

Mathematische Nachrichten > 284 > 5‐6 > 639 - 652

Using a multiple critical points theorem for non‐differentiable functionals, we investigate the existence and multiplicity of solutions for p(x)‐Laplacian Dirichlet problems with discontinuous nonlinearities. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

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Search results

Nonlinear boundary value problems of a class of elliptic equations involving critical variable exponents

New characterizations of Musielak-Orlicz-Sobolev spaces via sharp ball averaging functions

Multiplicity of solutions for Schrödinger systems with critical exponent via Ljusternik–Schnirelman theory

An integro-differential inequality related to the smallest positive eigenvalue ofp(x)-Laplacian Dirichlet problem

A minimization problem with variable growth on Nehari manifold

Density of smooth functions in variable exponent Sobolev spaces

Existence of three solutions for variable exponent elliptic systems

Higher integrability for nonlinear elliptic equations with variable growth and discontinuous coefficients

Existence of a nontrival solution for Dirichlet problem involving p(x)-Laplacian

Solutions for nonlinear elliptic equations with variable growth and degenerate coercivity

Higher integrability for nonlinear elliptic equations with variable growth

Variable Exponent Sobolev Spaces for Semilinear Elliptic Systems

Ni-Serrin type equations arising from capillarity phenomena with non-standard growth

Infinitely many homoclinic solutions for nonautonomous p(t)-Laplacian Hamiltonian systems

Multiplicity of Solutions on a Nonlinear Eigenvalue Problem for p(x)-Laplacian-like Operators

Existence result for hemivariational inequality involving p(x)-Laplacian

Existence Result for Diﬀerential Inclusion with p(x) - Laplacian

Existence of periodic solutions for a differential inclusion systems involving the p(t)-Laplacian

Existence of at least five solutions for a differential inclusion problem involving the p(x)-Laplacian

Discontinuous elliptic problems involving the p(x)‐Laplacian

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