The Dirichlet problem in an unbounded cone-like domain for second order elliptic quasilinear equations with variable nonlinearity exponent

. In this paper we consider the Dirichlet problem for quasi-linear second-order elliptic equation with the m ( x ) -Laplacian and the strong nonlinearity on the right side in an unbounded cone-like domain. We study the behavior of weak solutions to the problem at infinity and we find the sharp exponent of the solution decreasing rate. We show that the exponent is related to the least eigenvalue of the eigenvalue problem for the Laplace–Beltrami operator on the unit sphere.


Introduction
In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent, see e.g.[4, 16, 17, 21-23, 28, 29] and references therein.The basic properties of variable exponent function spaces were derived by O. Kováčik and J. Rákosník in [18] and (by different methods) by X.-L.Fan and D. Zhao in [14].For a comprehensive survey concerning Lebesgue and Sobolev spaces with variable exponent we refer to [12].
Differential equations and variational problems with m(x)-growth conditions arise from the study of elastic mechanics, oscillation problem, electrorheological fluids [11,24,25], image restoration [10], thermistor problem [31] and other.Moreover, the motion of a compressible fluid in a nonhomogeneous anisotropic porous medium obeys to nonlinear the Darcy law [3].The model of electrorheological fluids considered in [25] includes an integral of the symmetric part of gradient in a variable power which is caused by the action of an electromagnetic field.A similar structure of energy is also presented in the thermorheological model proposed in [30] for fluids with the stress tensor depending on the temperature.This system can be referred to as a coupled Boussinesq type sytem for a non-Newtonian fluid.
Corresponding author.Email: dawi@matman.uwm.edu.plOur interest is in the studying of the behavior of weak solutions to the Dirichlet problem with boundary condition on the lateral surface of a cone-like unbounded domain at infinity.For other results in unbounded and bounded cone-like domains we refer to [5][6][7][8]27].We refer also to some very recent works dealing with complementary aspects [20,26].These works can provide some ideas for further investigations in the cone-like domain too.For putting more emphasis on the effects of a gradient dependent reaction in the principal equation we refer to [15,19].
This paper is organized as follows.At first, we formulate the Dirichlet problem in an unbounded cone-like domain for second order elliptic quasilinear equations with variable nonlinearity exponent.Then, we introduce notations and function spaces that are used in the following sections.The main result, Theorem 1.2, is also formulated.In Section 2 we formulate an eigenvalue problem for the Laplace-Beltrami operator on the unit sphere, a Friedrichs-Wirtinger type inequality and some auxiliary inequalities and lemmas.In the next sections local estimate of the weighted Dirichlet integral and local estimate of weak solutions at infinity are investigated.Finally in Section 5 the power modulus of continuity near the infinity for weak solutions is considered.
Let B 1 (O) be the unit ball in R n , n ≥ 2 with center at the origin O and G ⊂ R n \ B 1 (O) be an unbounded domain with the smooth boundary ∂G.We assume that (QL) The following conditions will be needed throughout the paper: (ii) the function m(x) is Hölder continuous in G R , i.e. there exist a positive constant M and an exponent α ∈ (0, 1) such that where m(+∞) = lim |x|→+∞ m(x) = 2; We introduce the following notations: • C : a rotational cone {x 1 > r cos ω 0 2 }; • ∂C : the lateral surface of C : {x 1 = r cos ω 0 2 }; • Ω : a domain on the unit sphere S n−1 with smooth boundary ∂Ω obtained by the intersection of the cone C with the sphere S n−1 ; and the class of functions is the Sobolev space of those functions with zero trace on Γ R that, together with all their first order distributional derivatives, are L 1 -integrable in G R .
We denote We use the Sobolev embedding theorem for functions φ ∈ W 1,q 0 G 2 1 : where x ′ = 1 ϱ x, ϱ > R. Our main theorem is the following: Theorem 1.2.Let u be a weak solution of problem (QL), l = max m(x) : x ∈ G 2ϱ ϱ , λ − be as in (2.4) and assumption (i)-(iv) be satisfied.Then there exist R ≫ 1 and a positive constant C such that 2 Preliminaries

Eigenvalue problem
We consider the eigenvalue problem for the Laplace-Beltrami operator ∆ ω on the unit sphere which consists of the determination of all values ϑ (eigenvalues) for which (EVP) has non-zero weak solutions ψ(ω) ̸ = 0 (eigenfunctions).
Definition 2.1.A function ψ is said to be a weak solution of problem (EVP) provided that ψ ∈ W 1 0 (Ω) and satisfies the integral identity Throughout the paper we need only the least positive eigenvalue: For the existence problem of the least positive eigenvalue to problem (EVP) see for example Section 8.2.3 [9].

The Friedrichs-Wirtinger type inequality
From the definition of ϑ * (Ω) we obtain the following Friedrichs-Wirtinger type inequality: holds with the sharp constant 1 ϑ * .
Then for any ϱ > R and for all α provided that the integral on the right is finite.
Lemma 2.6.Let u(x) be a weak solution of (QL) and assumptions (i)-(iv) hold.Then We consider the solution u to the problem (QL) in the domain is Hölder continuous at infinity.First of all, by the mean value Lagrange theorem, we have where t is a negative number between m − − m(ϱx ′ ) and m − − m(+∞).Hence and by the Hölder assumption (ii), we get Now, using first derivative test, we can conclude that Thus, we obtain the required Further, assumptions (i), (iii) yield: and therefore we can apply the X.Fan Theorem 1.2 and Remark 5.2 [13] about a priori estimate of the gradient modulus of the problem (QL ′ ) solution max Returning to variable x and function u(x), we obtain , ϱ > R.
Lemma 2.7.Let u be a weak solution of problem (QL) and assumptions (i)-(iv) be satisfied.Then we have: Proof.At first we will show the convergence of the first integral.We set We choose η = uη m + k as a test function in (I I).Then we obtain: Next, using the Young inequality, with q = m(x) m(x)−1 , q ′ = m(x), we get Thus, from (2.9) we get Next, using assumption (iii), the inequality above yields In view of the choice of η k , we get We use the fact [2] that any solution u is Hölder continuous in G R : |u| Hence, by assumption (ii) and because we can estimate (2.12) In this way, from (2.10) Multiplying both sides of (2.14) by r 2−n k , by the definition of r k , we find Summing up above inequalities for all k = 0, 1, 2, . . ., we obtain Thus, the convergence of the first integral in (2.7) is proved.Now we observe that, in virtue of (2.5), (ii) and (2.11), we get which, by (2.15), yields the convergence of the second integral in (2.7).
We shall prove (2.8).Applying the Young inequality with q = m(x) m(x)−1 , q ′ = m(x) we have We can estimate the first integral using (2.5) and (2.12) in the following way: while the second integral using (2.13): From above inequalities we get lim ≤ lim which is the required (2.8).
We indicate another consequence of the integral identity (I I) for solutions u to the problem (QL) which is essentially used in the further consideration.

Lemma 2.8. If assumptions
and take a test function η(x) = r 2−n η N (r)u(x) in the integral identity (I I).Calculating we arrive at the equality .
First of all we observe that by assumption (iii) we have ub(x, u, u x ) ≤ µ|∇u| m(x) .In virtue of (2.7) it is clearly that lim Applying now the Young inequality with q = m(x) m(x)−1 , q ′ = m(x) we have by (2.13) and (2.15).Consequently lim .

Now we consider the integral
and hence, by (2.8) lim Next, because of (2.18), lim and therefore we can apply the L'Hospital rule: and lim Hence lim

Local estimate of the weighted Dirichlet integral
Theorem 3.1.Let u be a weak solution of problem (QL) and assumptions (i)-(iv) be satisfied.Let λ − be as in (2.4).Then there exist R ≫ 1 and a constant C > 0 such that Proof.We rewrite the inequality (2.16) in the form: Now, we observe that In fact, we get Passing to the limit N → +∞ we obtain (3.2).
By assumption (iii), we get Hence and from (3.1), (3.2) it follows that Let us estimate the integrals: To estimate them we set where the constant γ < −1 will be defined above.
By assumption (ii) and (2.11) for any x ∈ F 1 , we get In this way Next, (ii) yields for x ∈ F 2 , that Hence, once again in virtue of (ii) and by the inequality Applying inequality (2.6) with δ = − α 2γ , we get Eventually, we find that Integrals I 2 and I 3 are estimated similarly.Arguing as in (3.4), (3.5), we establish that |∇u| From (3.9) and by our assumption about Hölder continuity we get on the set F 2 .Thus in virtue of the Hardy-Wirtinger inequality (2.2), where ϑ * is the smallest positive eigenvalue of the Dirichlet problem for the Laplace-Beltrami operator in the domain Ω.Using (3.11), we obtain the estimate Now, by (3.13), we have Taking into account (3.12) we find that Applying the Cauchy inequality and (2.1), we have In this way we have the Cauchy problem (CP) with Now we show that U(R) ≤ U 0 = const.We can rewrite inequality (3.16) in the following form Since γ < −1 for sufficiently large ϱ ≥ 1, we have Now, by assumption (iii) regarding that |v| < v and in virtue of ϱ −m(ϱx ′ ) ≥ ϱ −l we obtain Next, by assumption (ii) we can estimate for all This estimation, with regard to (2.11) implies that For estimating the integral from the right-hand side of (4.1), we apply the Young inequality with p = m(ϱx ′ ) m(ϱx ′ )−1 , q = m(ϱx ′ ), δ = δ l : Hence, (4.1) takes the following form: For any κ ∈ (1, 2) we define sets G ′ (j) ≡ G 2 κ−(κ−1)2 −j , j = 0, 1, 2, . . .We see at once that Now we consider the sequence of cut-off functions and the number sequence t j = t n j , j = 0, 1, 2, . . . .We rewrite the inequality (4. . is the unit sphere (see Figure1.1).