Existence of solutions for p(x)-Laplacian Dirichlet problem☆
Introduction
The study of variational problems with nonstandard growth conditions is an interesting topic in recent years. p(x)-growth conditions can be regarded as an important case of nonstandard (p,q)-growth conditions. Many results have been obtained on this kind of problems, for example [2], [3], [4], [6], [7], [9], [17], [18]. Under (p,q)-growth conditions Marcellini [9] proved the existence and regularity of weak solutions of elliptic equations of divergence form with differentiable coefficients in the nondegenerate case. Under p(x)-growth conditions, Fan and Zhao [4], [6], [7] proved that the weak solutions of elliptic equations are Hölder continuous and its gradients have high integrability.
In this paper, we consider the existence of weak solutions of the problemwhere is a bounded domain, 1<p(x), and . Our aim is to give several existence results of weak solutions for problem (P). These results are extensions of that of p-Laplacian problems.
This paper is divided into four sections. In the second section, we introduce some basic properties of the generalized Lebesgue–Sobolev spaces which can be regarded as a special class of generalized Orlicz–Sobolev spaces. In the third section, several important properties of p(x)-Laplace operator are presented. Finally, in the fourth section, we give some existence results of weak solutions of problem (P).
Section snippets
The spaces
In order to discuss problem (P), we need some theories on spaces which we call generalized Lebesgue–Sobolev spaces. Firstly we state some basic properties of spaces which will be used later (for details, see [5], [10], [16], [15]). WriteWe can introduce the norm on byand , |·|p(x)) becomes
Properties of p(x)-Laplace operator
In this section, we discuss the p(x)-Laplace operator −△p(x)u≔−div(|∇u|p(x)−2∇u). Consider the following functional:
We know that (see [1]), J∈C1(X,R), and the p(x)-Laplace operator is the derivative operator of J in the weak sense. We denote , then Theorem 3.1 (i) is a continuous, bounded and strictly monotone operator; (ii) L is a mapping of type (S+), i.e. if in X and then un→u in X;
Existence of solutions
In this section we will discuss the existence of weak solutions of (P). Definition 4.1 We call that is a weak solution of (P), if If f is independent of u, we have Theorem 4.2 If f(x,u)=f(x), , where satisfies then (P) has a unique weak solution. Proof According to Proposition 2.5(ii), (for any v∈X) defines a continuous linear functional on X. Since L is a homeomorphism, (P) has a unique solution. Hereafter, f(x,t) is
References (19)
Regularity of nonstandard Lagrangians f(x,ξ )
Nonlinear Anal.
(1996)- et al.
A class of De Giorgi type and Hölder continuity
Nonlinear Anal.
(1999) - et al.
The quasi-minimizer of integral functionals with m(x) growth conditions
Nonlinear Anal.
(2000) Critical Point Theory and Applications
(1986)The regularity of Lagrangians f(x,ξ )=|ξ |α(x) with Hölder exponents α (x)
Acta Math. Sinica, New Ser.
(1996)- et al.
Regularity of minimizers of variational integrals with continuous p(x)-growth conditions
Chinese Ann. Math.
(1996) - et al.
On the generalized Orlicz-Sobolev space
J. Gansu Educ. College
(1998) - et al.
Singular solutions of the p-Laplace equation
Math. Ann.
(1985) Regularity and existence of solutions of elliptic equations with (p,q)-growth conditions
(1991)
Cited by (796)
Modular uniform convexity structures and applications to boundary value problems with non-standard growth
2024, Journal of Mathematical Analysis and ApplicationsOn eigenvalue problems for the p(x)-Laplacian
2024, Journal of Mathematical Analysis and ApplicationsStable critical points of even functionals on the Banach space and applications
2024, Journal of Mathematical Analysis and ApplicationsMULTIPLE SOLUTIONS FOR <![CDATA[ $p(x)$ ]]> -LAPLACIAN EQUATIONS WITH NONLINEARITY SUBLINEAR AT ZERO
2024, Bulletin of the Australian Mathematical SocietyGround state solutions for a kind of superlinear elliptic equations with variable exponent
2024, Boundary Value ProblemsMULTIPLICITY OF SOLUTIONS FOR FRACTIONAL κ(X)-LAPLACIAN EQUATIONS
2024, Journal of Applied Analysis and Computation
- ☆
Research supported by the National Science Foundation of China (19971036) and the Natural Science Foundation of Gansu Province (ZS991-A25-005-Z).