Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz—Sobolev space is discussed. Adams, Sobolev Spaces, Pure Appl. Afrouzi, A. Hadjian and G.

Molica Bisci, Some remarks for one-dimensional mean curvature problems through a local minimization principle, Adv. Nonlinear Anal. Afrouzi, S. Heidarkhani and S. Moradi, Existence of weak solutions for three-point boundary-value problems of Kirchhoff-type, Electron.

Differential EquationsPaper No. Shokooh, Infinitely many solutions for Steklov problems associated to non-homogeneous differential operators through Orlicz—Sobolev spaces, Complex Var. Elliptic Equ. Bohner, G. Caristi, S.

### Some existence results for a new class of elliptic Kirchhoff equation with logarithmic source terms

Moradi, A critical point approach to boundary value problems on the real line, Appl. Heidarkhani and A. Salari, Three solutions for a class of nonhomogeneous nonlocal systems: An Orlicz—Sobolev space setting, Dynamic Syst.

Bonanno, Relations between the mountain pass theorem and local minima, Adv. Bonanno and G. Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound.

Icable di perletti paolo in grumello del monteValue Probl.Metrics details. In this note, we study the existence and multiplicity of solutions for the quasilinear elliptic problem as follows:. The existence and multiplicity of solutions are obtained by a version of the symmetric mountain pass theorem. In this note, we discuss the existence and multiplicity of solutions of the following boundary value problem:. There is a large number of papers on the existence of solutions for the p-Laplacian equation. But the problem 1. For example, it is inhomogeneous and has an important physical background, e.

So, in the discussions, some special techniques are needed, and the problem 1. In paper [ 9 ], Fang and Tan discussed the problem 1. Motivated by their results, in this note, we discuss the problem 1. The paper is organized as follows. Obviously, the problem 1. To deal with this situation, we introduce an Orlicz-Sobolev space setting for the problem 1.

Now, we will make the following assumptions on f xt. Lemma 2. Using the version of the symmetric mountain pass theorem mentioned above, we can state our result as follows. Theorem 2. Lemma 3. Proof We prove the lemma by contradiction.

This proves the lemma. Now, it suffices to verify that. The proof is complete. Partial Differ. Ekvacioj— Fukagai N, Narukawa K: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Pura Appl. Nonlinear Differ. Nonlinear Anal.The paper deals with the study of the existence of weak positive solutions for a new class of the system of elliptic differential equations with respect to the symmetry conditions and the right hand side which has been defined as multiplication of two separate functions by using the sub-supersolutions method Mathematics Subject Classification: 35J60, 35B30, and 35B Elliptic systems of differential equations are of crucial importance in modelization and description of a wide variety of phenomena such as fluid dynamics, quantum physics, sound, heat, electrostatics, diffusion, gravitation, chemistry, biology, simulation of airplane, calculator charts, and time prediction.

PDEs are equations involving functions of several variables and their derivatives and model multidimensional systems generalizing ODEs ordinary differential equationswhich deal with functions of a single variable and their derivatives see, for example, [ 1 — 15 ]. Malgrange—Ehrenpreis theorem states that linear partial differential equations with constant coefficients always have at least one solution; another powerful and general result in case of polynomial coefficients is the Cauchy—Kovalevskaya theorem ensuring the existence and uniqueness of a locally analytic solution for PDEs with coefficients that are analytic in the unknown function and its derivatives; otherwise, the existence of solutions is not guaranteed at all for nonanalytic coefficients even if they have derivatives of all orders see [ 16 ].

Given the rich variety of PDEs, there is no general theory of solvability. Instead, research focuses on particular PDEs that are important for applications. It would be desirable when solving a PDE to prove the existence and uniqueness of a regular solution that depends on the initial data given in the problem, but perhaps we are asking too much.

A solution with enough smoothness is called a classical solution, but in most cases as for conservation laws, we cannot achieve that much and allow generalized or weak solutions. The point is this: looking for weak solutions allows us to investigate a larger class of candidates, so it is more reasonable to consider as separate the existence and the regularity problems.

For various PDEs, this is the best that can be done, and naturally nonlinear equations are more difficult than linear ones. Overall, we know too much about linear PDEs and in best cases, we can express their solutions but too little about nonlinear equations.

For linear PDEs, various methods and techniques can be used for separation of variables, method of characteristics, integral transform, change of variables, superposition principle, or even finding a fundamental solution and taking a convolution product to obtain the solution.

Variational theory is the most accessible and useful of the methods for nonlinear PDEs, but there are other nonvariational techniques of use for nonlinear elliptic and parabolic PDEs such as monotonicity and fixed point methods, semigroup theory, and sub-supersolutions method that played an important role in the study of nonlinear boundary value problems for a long time. Then, the SSM were also used to study Dirichlet and Neumann boundary value problems for semilinear elliptic problems in [ 2324 ], and even for nonlinear boundary value problems in [ 25 — 27 ] and also for systems of nonlinear ordinary differential equations in [ 28 — 30 ].

The concept of weak sub-supersolutions and supersolutions was first formulated by Hess and Deuel in [ 3132 ] to obtain existence results for weak solutions of semilinear elliptic Dirichlet problems and was subsequently continued by several authors see, e.

The study of differential equations and variational problems with nonstandard - growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc see [ 44 ]. Many existence results have been obtained on this kind of problems see, for example, [ 44 — 57 ] and in [ 45 ] a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term has been investigated.

In the last few years in [ 5158 — 60 ], the regularity and existence of solutions for differential equations with nonstandard - growth conditions have been studied and - Laplacian elliptic systems with a constant have been archived. In this work, we study the existence of weak positive solutions for a new class of the system of differential equations with respect to the symmetry conditions by using sub-supersolution method.

### On a Class of Anisotropic Nonlinear Elliptic Equations with Variable Exponent

In this paper, we consider the system of differential equations: where is a bounded smooth domain with boundary and are functions with, and is a - Laplacian defined as and are continuous functions, while are monotone functions in such that satisfying some natural growth condition at. We point out that the extension from - Laplace operator to - Laplace operator is not trivial, since the - Laplacian has a more complicated structure then the - Laplace operator, such as it is nonhomogeneous.

Moreover, many results and methods for - Laplacians are not valid for the - Laplacian; for example, if is bounded, then the Rayleigh quotient is zero in general, and only under some special conditions, is positive see [ 53 ]. Maybe the first eigenvalue and the first eigenfunction of the - Laplacian do not exist, but the fact that the first eigenvalue is positive and the existence of the first eigenfunction are very important in the study of - Laplacian problem.

There are more difficulties in discussing the existence of solutions of variable exponent problems. In [ 59 ], the authors considered the existence of positive weak solutions for the following - Laplacian problem: where the first eigenfunction has been used to construct the subsolution of - Laplacian problem.

In [ 48 ], the existence and nonexistence of positive weak solutions to the following quasilinear elliptic system: has been considered where the first eigenfunction has been used to construct the subsolution of problem 7 and the following results were obtained: i Ifthen problem 7 has a positive weak solution for each.We are concerned with the existence of infinitely many radial symmetric solutions for a nonlinear stationary problem driven by a new class of nonhomogeneous differential operators.

The proof relies on the symmetric version of the mountain pass theorem. RabinowitzDual variational methods in critical point theory and applications, J. Functional Anal. Google Scholar. AzzolliniMinimum action solutions for a quasilinear equation, J. Partial Differential Equations49 MingioneNon-autonomous functionals, borderline cases and related function classes, St.

Petersburg Mathematical Journal27 BrowderPartial differential equations in the 20th century, Adv. MingioneBounded minimisers of double phase variational integrals, Archive for Rational Mechanics and Analysis, Partial Differential Equations33 HalseyElectrorheological fluids, Science, Jabri, The Mountain Pass Theorem. KimMountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents, Manuscripta Math.

Kristaly, V. MarcelliniRegularity and existence of solutions of elliptic equations with pq -growth conditions, J.

Differential Equations90 PalaisThe principle of symmetric criticality, Commun. Pucci and V. Unione Mat.This paper examines a class of nonlocal equations involving the fractional p -Laplacian, where the nonlinear term is assumed to have exponential growth.

More specifically, by using a suitable Trudinger—Moser inequality for fractional Sobolev spaces, we establish the existence of weak solutions for these equations. Download to read the full article text. Bartsch T. Partial Differ. Brezis, H. Springer, New York Caffarelli L. Nonlinear Partial Differ. Abel Symp.

Carl S. Cheng, M. A 44 Mathematische Nachrichten— Di Nezza E. Ding Y. Guo B. Iannizzotto A. Kozono H. Indiana Univ. Lam N. Laskin N. A— Laskin, N. E 66 Lindgren E. Moser J. Nelson E.

## A Priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations

Ozawa T.Djairo G. Unable to display preview. Download preview PDF. Skip to main content. This service is more advanced with JavaScript available. Advertisement Hide. Authors Authors and affiliations D. Lions R. Chapter First Online: 08 January This is a preview of subscription content, log in to check access.

A gmonA. D ouglis and L. Pure Appl.

Google Scholar. A mannFixed point equations and nonlinear eigenvalue problems in ordered Banach spaces S. A mbrosetti and P. Functional AnalysisVol. Royal Soc. B erestycki and P. Analyse MathematiqueVol. B rezis and T. K ato. B rezis and R. C randall and P.We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneous -harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equation in the weighted variable exponent Sobolev space.

In [ 1 — 5 ], the nonhomogeneous -harmonic equation for differential forms has received much investigation. In [ 6 ], the obstacle problem of the -harmonic equation for differential forms has been discussed. However, most of these results are developed in the space or space. Meanwhile, in the past few years the subject of variable exponent space has undergone a vast development; see [ 7 — 11 ].

Best price on super cleanFor example, [ 8 — 10 ] discuss the weighted and spaces and the weak solution for obstacle problem with variable growth has been studied in [ 1011 ]. In this paper, we are interested in the following obstacle problem: for belonging to where,; means that, for anywe haveand the variable exponent satisfies. The operators and satisfy the following growth conditions on a bounded domain : H1 and are measurable for all with respect to and continuous for with respect toH2H3H4H5H6 for. We will discuss the existence and uniqueness of the solution for the abovementioned obstacle problem.

**Heat equation: Separation of variables**

Throughout this paper, we assume that is a bounded domain in. Let be the set of all -forms in.

Tcpdump javaA differential -form is generated by ; that is,where is differential function,and. Let be the space of all differential -forms on. For andthen the inner product is obtained by. We write. We denote the exterior derivative by for. Its formal adjoint operator is given by; here is the well-known Hodge star operator. Denote the class of infinitely differential -forms on by. A differential -form is called a closed form if in. Next we will introduce some basic properties of weighted variable exponent Lebesgue spaces and weighted variable exponent Sobolev spacesand we define to be the set of all n -dimensioned Lebesgue measurable functions.

Functions are called variable exponents on. We define. Ifthen we call a bounded variable exponent.

How to select all rows in datagridview on right mouse clickIfthen we define bywhere. The function is called the dual variable exponent of.

We denote as a weight if and a. From [ 710 ], we know that if satisfies 3the weighted variable exponent Lebesgue spaces with the norm and the weighted variable exponent Sobolev spaces with the norm are Banach space and reflexive and uniformly convex. On the set of all differential forms onwe define the weighted variable exponent Lebesgue spaces of differential -forms and the weighted variable exponent Sobolev spaces of differential forms. Definition 1. We denote the weighted variable exponent Lebesgue spaces of differential -forms by and we endow with the following norm: And the spaces with the norm are the weighted variable exponent Sobolev spaces of differential -forms.

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