Existence of radial solutions for a p ( x ) -Laplacian Dirichlet problem

In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized p ( x )-Laplacian problem with Dirichlet boundary condition in the unit ball in R N (for N ≥ 3), where a , b , R are radial functions.


Introduction
The study of differential equations and variational problems with nonstandard p(x)growth conditions (or nonstandard (p, q)-growth conditions) is an attractive topic and has been the object of considerable attention in recent years (see [1]). The reasons for such an interest are as follows: 1) Physically, it relies on the fact that they model phenomena arising from various fields such as the motion of electrorheological fluids, which are characterized by their ability to drastically change their mechanical properties under the influence of an exterior electromagnetic field, the thermo-convective flows of non-Newtonian fluids, and the image processing; 2) Mathematically, it relies on the fact that the standard mathematical techniques are not adequate to study these problems and they need new techniques. This may be the central development of mathematical ideas in active areas of pure mathematics, which have had a decisive interaction with PDEs (such as [2][3][4][5][6][7][8][9][10][11][12][13][14]).
The aim of this paper is to prove the existence of at least one positive radial solution belonging to the space W where θ , ξ ∈ L ∞ (0, 1) are such that θ is a positive nonconstant radially nondecreasing function, and ξ is a nonnegative radially nonincreasing function. The statement of the main result of this paper is as follows.
Theorem 1.1 Let B be the unit ball centered at the origin in R N , N ≥ 3, and let p, q, r ∈ C + (B) be such that p + , are nonconstant radial functions as in (2). The p(x)-Laplacian Dirichlet problem (1) admits at least one radial Before verifying our approach, we prepare some preliminaries. From now on we assume that B is the unit ball centered at the origin in R N , N ≥ 3, and we set where p ∈ C + (B) = {g ∈ C(B) : g -> 1}. The generalized Lebesgue space L p(·) (B) is the collection of all measurable functions u on B such that B |u(x)| p(x) dx < +∞ with the norm For any u ∈ L p(·) (B) and v ∈ L p (·) (B), where L p (·) (B) is the conjugate space of L p(·) (B), we have the Hölder-type inequality The following proposition is well known in Lebesgue spaces with variational exponent (e.g., see [15,Proposition 2.7]).

Theorem 1.3
Let be a bounded smooth set in R N , and let p, q ∈ C + (¯ ). Then As a consequence of Theorem 1.3, we have if p(x) ≤ q(x) for a.e. x ∈ .
Now we recall some notations and results to be used further. For radial functions a, b ∈ L ∞ (B) given by (2), we consider the spaces Definition 1.7 (Subdifferential) Let V be a real Banach space, and let V * be its topological dual with pairing between V and V * denoted by ·, · . Let : V → (-∞, +∞] be a proper convex function, and let 2 V * be the set of all subsets of V * . The subdifferential ∂ : V → 2 V * of is defined as the following set-valued operator: Note that if is Gâteaux differentiable at u with its derivative denoted by D (u), then ∂ (u) is a singleton. In this case, ∂ (u) = {D (u)}.  : V → (-∞, +∞] be a proper (i.e., Dom = ∅), convex, and lower semicontinuous function. Let K ⊂ V be a weakly convex closed set. Define the function K : V → (-∞, +∞] by Consider the functional We say that u ∈ V is a critical point of I K if D (u) ∈ ∂ K (u) or, equivalently, it satisfies in the inequality Note that a global minimum point is a critical point.

Definition 1.11 (PS compactness condition)
We say that I K (12) satisfies the Palais-Smale (PS) compactness condition if for any sequence {u n } such that for all v ∈ V as n → 0, then {u n } possesses a convergent subsequence.
The following mountain pass geometry (MPG) theorem was proved in [20].  Proof Let u := lim n→∞ u n , and let ε > 0. We have to show that for n large enough, Since u is assumed to be continuous, it is uniformly continuous on the compact interval Since |u n (x i )u(x i )| and |u n (x i+1 )u(x i+1 )| are not greater than ε 2 , it follows that Moreover, since the oscillation of u on [x i , x i+1 ] is less than ε 2 and since x ∈ [x i , x i+1 ], we also have u(x i ) ≥ u(x) -ε 2 and u(x i+1 ) ≤ u(x) + ε 2 . Altogether, this gives which concludes the proof.

Definition 2.3 (Weak solution)
Let p, q, r ∈ C + (B), a, b ∈ L ∞ (B), and R be given as in Theorem 1.1, and let V be the space as in (14). We say that u ∈ V is a (weak) solution of problem (1) if u is increasing and satisfies the Dirichlet boundary conditions and also if the following equality is true for all w ∈ V : To prove the claim, we consider the Euler-Lagrange energy functional corresponding to problem (1) over the convex closed set Concerning Theorem 2.2, we define ψ, ϕ : V → R by and Notice that ψ is a proper, convex, and lower semicontinuous and Dϕ(u) = a(x)|u| q-2 u. Therefore ϕ is a C 1 -function on the space V . Let us introduce the functional I K : V → (-∞, +∞] defined by where ψ K is as in (11). Note that I K = I = ψϕ on K . We prove Theorem 1.1 in two steps.
Step 1. We show that I K = ψ Kϕ has a critical point in K . For this reason, we use the MPG theorem (Theorem 1.12), in which we will need the following lemma.
Proof Suppose that {u n } is a sequence in K such that I K (u n ) → c ∈ R, n → 0, and (20) for all v ∈ V . We show that {u n } has a convergent subsequence in V . First, notice that u n ∈ Dom(ψ). Then Thus, for large values of n, we have Furthermore, Now consider the function g(s) = s r +q -(s -1)-1 on the interval (1, +∞). Set s * = ( qr + ) 1 r + -1 . Obviously, g(s) < 0 for all s ∈ (1, s * ). We choose such a number s. So we have s > 1 and s r + -1 < q -(s -1). Setting v = su n in (20), we can see that Therefore (s -1)qϕ(u n )s r + -1 ψ K (u n ) ≤ n (s -1) u n V ≤ C n u n V .
Multiplying (24) by α and summing it up with (21), we get where is defined as in (4). Then So by Lemma (2.4) there exists C > 0 such that u n p * ≤ C 1 + u n * .
Proof We show that I satisfies the conditions of the MPG theorem. It is clear that I(0) = 0. Take e ∈ K . Then it follows that Now let p + , r + < q -. Then for t large enough, I(te) is negative. Condition (iii) of the MPG theorem is satisfied. Let u ∈ Dom(ψ) with u V = ρ > 0. Notice that by Lemma 2.4, for u ∈ K , we have Also, where C is the same as in Proposition 1.4(ii), and q = ⎧ ⎨ ⎩ q -, |u| p(·) < 1, q + , |u| p(·) ≥ 1.
Step 2. We show that the triple (ψ K , ϕ, K) satisfies the pointwise invariance condition at u when G = 0. To show this statement, we need following lemma.
admits at least one weak solution.
Proof First, notice that by integration respect to u we can see that there exist α 1 , β 1 > 0 such that By the Hölder inequality, Remark 1.5, and Lemma 1.6 we have J(u) ≥ 1 p + u p * -C u * .