Existence and multiplicity of solutions for p(x)p(x)-Laplacian equations in RNRN☆

Infinitely many solutions for elliptic problems with variable exponent and nonlinear boundary conditions

“…(Note that, since [28] is written in Chinese, the above assertions are based on the affirmations made by Fan and Han in [9], or by Yao in [27].) For k = 1, 2, .…”

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR AN ELLIPTIC EQUATION WITH $p(x)$-GROWTH CONDITIONS

“…By Lemma 1 (i) it follows that on the boundary of the unit ball, centered at the origin in E and denoted by B 1 (0), we have inf On the other hand, by Lemma 1 (iii) there exists ϕ ∈ E such that J(tϕ) < 0, for all t > 0 small enough. Moreover, relations (8), (9) and (4) imply that, for any u ∈ B 1 (0), the inequality…”

“…More precisely, such equations are used to model phenomena which arise in elastic mechanics or electrorheological fluids. For a general account of the underlying physics and for some technical applications we refer to [ [3,9,11].) However, we point out that such generalizations are not trivial since the p(x)-Laplace operator possesses more complicated nonlinearity; for example it is inhomogeneous.…”

Existence of solutions for an elliptic p(x)-Kirchhoff-type systems in unbounded domain

“…More precisely, Kirchhoff proposed the following model 2) have the following meanings: E is the Young modulus of the material, ρ is the mass density, L is the length of the string, h is the area of cross-section, and P 0 is the initial tension. The study of elliptic problems involving p(x)−Laplacian has interested in recent years, for the existence of solutions see [1], [9] and [12], and the eigenvalue involving p (x) −Laplacian problems see [10] and [11].…”