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formerly Central European Journal of Mathematics

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Volume 15, Issue 1 (May 2017)

Issues

Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities

Mariusz Bodzioch
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  • Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Sloneczna 54, 10-710 Olsztyn, Poland
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/ Damian Wiśniewski
  • Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Sloneczna 54, 10-710 Olsztyn, Poland
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  • De Gruyter OnlineGoogle Scholar
/ Krzysztof Żyjewski
  • Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. Sloneczna 54, 10-710 Olsztyn, Poland
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Published Online: 2017-05-23 | DOI: https://doi.org/10.1515/math-2017-0033

Abstract

We have investigated the behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities in bounded and unbounded domains. We found exponents of the solution’s decreasing rate near the boundary singularities.

Keywords: Elliptic equations; Quasi-linear problems; Boundary singularities

MSC 2010: 35J20; 35J25; 35J60; 35J70

1 Introduction

This is a brief description of results of [13] with comments and improvements, which were presented at the International conference on differential equations dedicated to the 110th anniversary of Ya. B. Lopatynsky. In these articles we have investigated the behaviour of solutions of quasi-linear elliptic problems in a neighborhood of boundary singularities in bounded and unbounded domains. We have found exponents of the solution’s decreasing rate of the type |u(x)| ≤ O(|x|α), near the boundary singularities.

Let G ⊂ ℝn be an unbounded domain (see Fig. 1) with boundary ∂G that is a smooth surface everywhere except at the origin 𝓞 and near 𝓞 it is a conical surface, n ≥ 2. We assume that G=G0RGR, where G0R=G0dGdR and G0d is a rotational cone with the vertex at 𝓞 and the aperture ω0 ∈ (0,π), d ≪ 1, GR = {x = (r, ω) ∈ ℝn|r ∈ (R, ∞), ω ∈ Ω ⊂ Sn − 1, n ≥ 2}, R ≫ 1, Sn − 1 is the unit sphere.

an unbounded cone-like domain.
Fig. 1

an unbounded cone-like domain.

We introduce the following notations for a domain G which has a boundary conical point:

  • Ω : = GSn − 1, where Sn − 1 denotes the unit sphere in ℝn;

  • Ω: the boundary of Ω;

  • Gab := G ∩{(r, ω) : 0 ≤ a < r < b, ω ∈ Ω}: a layer in ℝn;

  • Γab := ∂G ∩{(r, ω) : 0 ≤ a < r < b, ωΩ}: the lateral surface of Gab;

  • Gd:=GG0d,Γd:=GΓ0d.

We use standard function spaces: Ck(G¯) with the norm |u|k,G¯;Lp(G) with the norm ∥up,G, p ≥ 1; the Sobolev space Wk,p(G) with the norm ∥uWk,p(G) for integer k ≥ 0; the weighted Sobolev-Kondratiev space Vp,αk(G) for integer k ≥ 0, 1 < p < ∞ and α ∈ ℝ with the finite norm uVp,αk(G)=(G|β|krα+p(|β|k)|Dβu|pdx)1p and the space Vp,αk,1p(Γ) of functions φ, given on ∂G, with the norm gVp,αk,1p(G)=infGVp,αk(G), where the infimum is taken over all functions g such that 𝓖|∂G = g in the sense of traces.

2 Oblique derivative problem

In [1] we have investigated the behaviour of strong solutions to the oblique derivative problem for the general second-order quasi-linear elliptic equation in a neighbourhood of a conical boundary point of an n-dimensional bounded domain, n ≥ 2. In the case of the linear equation we refer to [4, 5].

Local maximum principle for strong solutions to the elliptic quasi-linear oblique derivative problem in convex rotational cones has been obtained by Lieberman in [6]. He [7] and Trudinger [8] have obtained local gradient bound estimate and local Hölder gradient estimate of strong solutions in any sub-domain with a C2 boundary portion of the domain.

The results obtained in [1] are a generalization and improvement of results of [9] on the case of the oblique boundary condition. We consider the oblique derivative problem for the elliptic second-order linear equation: aij(x,u,ux)uxixj+a(x,u,ux)=0,aij=aji,xG0R,un+χ(ω)ur+1|x|γ(ω)u=g(x),xG0RO,(QL) where n denotes the unite exterior normal vector to G0RO,(r,ω) are spherical coordinates in ℝn with pole 𝓞; repeated indices are understood as summation from 1 to n.

Definition 2.1

A function u is called a strong solution of problem (QL) if uWloc2,q(G0RO)W1(G0R)C0(G0R¯),q>n and satisfies the equation for almost all xGεR for all ε > 0 as well as the boundary condition in the sense of traces on G0RO. We assume that M0=xmaxxG0R¯|u(x)| is known (see e.g. Theorem 13.1 [7]).

2.1 Assumptions

Let M={(x,u,z):xG0R,uR,zRn}. With regard to problem (QL) we assume that the following conditions are satisfied on 𝔐:

  1. aij(x,u,z)W1,q(M),q>n;a(x,u,z),a(x,u,z)u are Caratheodory functions;

  2. the uniform elipticity v|ξ|2<_i,j=1naij(x,u,z)ξiξj<_μ|ξ|2,ξRn, with the ellipticity constants μν>0;aij(0,0,0)=δij,i,j=1,,n -the Kronecker symbol;

  3. a(x,u,z)u0;

  4. γ(ω),χ(ω)C1(Ω¯); there exist numbers s>1,χ00,γ0>tanωO20 and γ1γ(ω) ≥ γ0 > 0, 0 ≤ χ(ω) ≤ χ0 as well as nonnegative constants μ1, k1, go, g1 and functions b(x),f(x)Llocq(G),qn such that the inequalities |a(x,u,z)|+|a(x,u,z)u|μ1|z|2+b(x)|z|+f(x),0b(x),f(x)k1|x|s2,|g(x)|g0|x|s1,|g|g1|x|s2, hold;

  5. coefficients of problem (QL) satisfy such conditions that guarantee uC1+x~(G) and the existence of the local a priori estimate |u|1+x~,GM1,x~(0,1), for any smooth G⊂⊂G0R¯O (see Theorems 13.13 and 13.14 [7]).

2.2 The main result

Theorem 2.2

([1]). Let u be a strong solution of problem (QL) and λ > 1 be the smallest positive eigenvalue of the following eigenvalue problem for the Laplace - Beltrami operator Δω on the unit sphere Δωψ+λ(λ+n2)ψ(ω)=0,ωΩ,ψν+λχ(ω)+γ(ω)ψ(ω)=0,ωΩ,(EVP) where ν is the unit exterior normal to G01 at the points of Ω. Suppose that assumptions (A) - (E) are satisfied. Then there exist numbers d > 0, c0, c1 independent of u such that

  • |u(x)|c0|x|λifs>λ,|x|λln321|x|ifs=λ,xG0d;|x|Sifs<λ,(1) |u(x)|c1|x|λ1ifs>λ,|x|λ1ln321|x|ifs=λ,xG0d.|x|s1ifs<λ,(2)

  • If 4 + q(λ − 2) > 0 then u(x)Vq,4n2(G0Q) and there exist numbers d > 0, c2, independent of u such that u(x)Vq,4n2(G0ϱ)C~2ϱλ2+4qifs>λ,ϱλ2+4qln31Qifs=λ,,0<ϱ<d,ϱs2+4qifs<λ,(3) and

  • if 1<λ<2,q>n2λ then u(x)Cλ(G0d¯)ifsλ,u(x)Cs(G0d¯)ifs<λ.(4)

Theorem 2.3

([4, 5]). There exists the smallest positive eigenvalue λ of problem (EVP), which satisfies the following inequalities 0<λ<(πω0)2+(n22)2n22 for n ≥ 3.

3 Quasi-linear nonlocal Robin problem

In [3] we have investigated the behaviour of weak solutions for the nonlocal Robin problem with quasi-linear elliptic divergence second-order equations in a plain domain in a neighbourhood of the boundary corner point 𝓞. In the case of the linear equation we refer to [10, 11].

Here, we consider a different eigenvalue problem (see (EVP2)) and derive a new Friedrichs-Wirtinger type inequality adapted to the quasi-linear elliptic problem considered in [3]. It allows us to improve the main result of [3] (see Theorem 3.7): the exponent of weak solution behavior (see inequality (20)) in a neighborhood of an angular point 𝓞 in the case 𝓑 < 0 is better than that one obtained in [3].

We consider the type of nonlocal problems, where the support of nonlocal terms intersects the boundary. Namely, the situation in which a part Γ of domain boundary ∂G is mapped by transformation γ on γ(Γ) and γ(Γ)¯G.

Let us consider the domain G0R ⊂ ℝ2. Moreover, let Γ+ and Γ be the part of boundary G0R for which x2 > 0 and x2 < 0 respectively. We assume, that G0R=Γ¯+Γ¯ is a smooth curve everywhere except at the origin 𝓞 ∈ G0R and near the point 𝓞 curves Γ± are lateral sides of an angle with the measure ω0 ∈ [0, π) and the vertex at 𝓞. Let Σ0 = G0R ∩{x2 = 0}, where OΣ0¯. Furthermore, let γ be a diffeomorphism mapping of Γ+ onto Σ0. Additionally, we suppose that there exists d > 0 such that in the neighbourhood of Γ0+d the mapping γ is the rotation about the origin 𝓞 and ω02 is the angle of rotation. It means γ(Γ0+d¯)=Σ0d¯=G0d¯Σ0.

We have considered a quasi-linear elliptic equation with the nonlocal boundary condition connecting the values of the unknown function u on the boudary part Γ+ with its values of u on Σ0 : ddxi(|u|q|u|m2uxi)+a0rmu|u|q+m2μu|u|q2|u|m=f(x),xG0R;|u|q|u|m2un+β+r1mu|u|q+m2+br1mu(γ(x))|u(γ(x))|q+m2=g(x,u),xΓ+;|u|q|u|m2un+βr1mu|u|q+m2=h(x,u),xΓ(QL2) here q ≥ 0, m > 1, μ ≥ 0, a0 ≥ 0, β± > 0, b ≥ 0 are given numbers and ndenotes the unit outward with respect to G0R normal to G0RO.

Recall that we are dealing with the nonlocal problem and the boundary G0R is non smooth, so the formulation of problem (QL2) does not make sense in general. To make our formulation precise, we give the definition of the weak solution of (QL2).

Definition 3.1

A function u is called a weak solution of problem (QL2) provided that uC0(G0R¯)Vm,01(G0R) and for all functions ηC0(G0R¯)Vm,01(G0R) the following integral identity G|u|q|u|m2uxiηxidx+G{a01rmu|u|q+m2μ|u|q1|u|msgnu}η(x)dx+β+Γ+1rm1u|u|q+m2η(x)ds+bΓ+1rm1u(γ(x))|u(γ(x))|q+m2η(x)ds+βΓ1rm1u|u|q+m2η(x)ds=Gf(x)η(x)dx+Γ+g(x,u)η(x)ds+Γh(x,u)η(x)ds is satisfied.

3.1 The Friedrichs-Wirtinger type inequality

We consider the following eigenvalue problem: (|ψ(ω)|m2ψ(ω))+ϑ|ψ(ω)|m2ψ(ω)=0,ωΩ|ψ(ω02)|m2ψ(ω02)+β+|ψ(ω02)|m2ψ(ω02)+b|ψ(0)|m2ψ(0)=0|ψ(ω02)|m2ψ(ω02)+β|ψ(ω02)|m2ψ(ω02)=0,(EVP2) with m ≥ 2, β+, β > 0, b ≥ 0, which consists in determining all values ϑ (eigenvalues) for which (EVP2) has nonzero weak solutions (eigenfunctions) ψ(ω).

Definition 3.2

A function ψ is called a weak solution of problem (EVP2) provided that ψW1,m(Ω) ∩ C0( Ω¯) and satisfies the integral identity Ω(|ψ(ω)|m2ψ(ω)η(ω)ϑ|ψ(ω)|m2ψ(ω)η(ω))dω+β+|ψ(ω02)|m2ψ(ω02)η(ω02)+β|ψ(ω02)|m2ψ(ω02)η(ω02)+b|ψ(0)|m2ψ(0)η(ω02)=0 for all η(ω) ∈ W1,m(Ω) ∩ C0( Ω¯i).

Theorem 3.3

(Friedrichs-Wirtinger’s type inequality). Let Ω ⊂ S be an arc, m ≥ 2, ϑ be the eigenvalue ofproblem (EVP2) and ψW1,m(Ω) ∩ C0( Ω¯) be the corresponding eigenfunction. Then for any uW1,m(Ω) ∩ C0( Ω¯), uconst ≠ 0 the inequality ϑΩ|u(ω)|mdωΩ|u(ω)|mdω+B|u(ω02)|m+β|u(ω02)|m(F − W) holds, where B=bψ(0)|ψ(0)|m2ψ(ω02)|ψ(ω02)|m+β+.(5)

Proof

Let us first derive the estimate (FW) for functions uC0( Ω¯) ∩ C1(Ω) and ψC1( Ω¯) ∩ C2(Ω). Setting u(ω) = ψ(ω)v(ω) we obtain |u(ω)|m=|(ψ(ω)v(ω))|m=[(v(ω)ψ(ω)+v(ω)ψ(ω))2]m2=[ψ2(ω)v2(ω)+2ψ(ω)ψ(ω)v(ω)v(ω)+ψ2(ω)v2(ω)]m2=|v(ω)|m|ψ(ω)|m[1+2ψ(ω)ψ(ω)v(ω)v(ω)+ψ2(ω)ψ2(ω)v2(ω)v2(ω)]m2.(6)

By Bernoulli inequality (1 + x)α ≥ 1 + αx, with α ≥ 1 and x > − 1 from (6) we have |u(ω)|m|v(ω)|m|ψ(ω)|m(1+m2[2v(ω)v(ω)ψ(ω)ψ(ω)+ψ2(ω)ψ2(ω)v2(ω)v2(ω)])|v(ω)|m|ψ(ω)|m+m|v(ω)|m2v(ω)v(ω)ψ(ω)ψ(ω)|ψ(ω)|m2.(7)

Next, in virtue of (|v(ω)|m)′ = m |v(ω)|m−2v(ω)v′(ω) and from (EVP2) it follows that: (|v(ω)|mψ(ω)ψ(ω)|ψ(ω)|m2)=(|v(ω)|m)ψ(ω)ψ(ω)|ψ(ω)|m2+|v(ω)|m|ψ(ω)|m+|v(ω)|mψ(ω)(|ψ(ω)|m2ψ(ω))=m|v(ω)|m2v(ω)v(ω)ψ(ω)ψ(ω)|ψ(ω)|m2+|v(ω)|m|ψ(ω)|m+|v(ω)|mψ(ω)(|ψ(ω)|m2ψ(ω))=|v(ω)|m|ψ(ω)|mm|v(ω)|m2v(ω)v(ω)ψ(ω)ψ(ω)|ψ(ω)|m2θ|v(ω)|m|ψ(ω)|m.(8) Now, from (7), (8) and definition of function u(ω) we get the inequality ϑ|u(ω)|m|u(ω)|m(|v(ω)|mψ(ω)ψ(ω)|ψ(ω)|m2).(9)

Integrating (9) over ω and since ψ(ω) is a solution of (EVP2) we obtain ϑΩ|u(ω)|mdωΩ|u(ω)|mdω|v(ω)|mψ(ω)ψ(ω)|ψ(ω)|m2|ω=ω02ω=ω02=Ω|u(ω)|mdω+β+|v(ω02)|m|ψ(ω02)|m+β|v(ω02)|m|ψ(ω02)|m+b|v(ω02)|mψ(ω02)ψ(0)|ψ(0)|m2=Ω|u(ω)|mdω+β++bψ(0)|ψ(0)|m2ψ(ω02)|ψ(ω02)|m|u(ω02)|m+β|u(ω02)|m. Thus, we get (FW) for smooths functions. The extension to arbitrary functions in W1,m(Ω)C0(Ω¯) follows by straight-forward approximation argument. □

Remark 3.4

In virtue of inequality (FW), Corollary 2.2 [3] for any vVm,01(G0d)C0(G¯0d) and ϱ ∈ (0, d), we have G0ϱ|v|mdxϱmϑ{G0ϱ|v|mdx+BΓ0+ϱ|v|mrm1ds+βΓ0ϱ|v|mrm1ds}(H − W) where ϑ is the least positive eigenvalue of (EVP2) problem.

3.2 Assumptions

With regard to problem (QL2) we assume that the following conditions are satisfied:

  1. let p > m > 1, 0μ<q+m1m1 be given numbers; g(x, u) be the Caratheodory and continuously differentiable with respect to variable u function Γ+ × ℝ → ℝ, h(x, u) be the Caratheodory and continuously differentiable with respect to variable u function Γ × ℝ → ℝ;

  2. h(x,u)u0,g(x,u)u0;

  3. f(x)Lpm(G0R);g(x,0),h(x,0)W1,pm1(G0R) and there exist positive constants g1, h1 such that |g(x,u)u|g1|u|m1,xΓ+;|h(x,u)u|h1|u|m1,xΓ;

  4. 0b<min{m(m1)m2[(m1)(1μ)+q](q+m1)mω0m1;mβ+1+(m1)2m1}.

For the following we will use the numbers: K:=min{ςm1(1μς)bm1mω0m1;1} with ς=m1q+m1,(10) K+:=min{K;1B(β+b1+(m1)2m1m)},B>0,(11) Ξ(m)=mm2(2m+2)m+22m,(12) kσ=:supϱ>0ϱmσ{G0ϱ|f(x)|mm1dx+Γ0+ϱ|g(x,0)|mm1ds+Γ0ϱ|h(x,0)|mm1ds},σ>1;(13) and ϑ~±:=K+ϑ1mΞ(m), if B>0Kϑ1mΞ(m), if B0, where 𝓑 is defined by (5) and ϑ is the smallest positive eigenvalue of problem (EVP2).

3.3 Integral estimates

Lemma 3.5

Let 𝓑 be defined by (5) and G0d be a sector. Let vC0(G0d¯)Vm,01(G0d) for almost all ϱ ∈ (0, d) and V±(ϱ)=G0ϱ|v|mdx+α±BΓ0+ϱ|v(x)|mrm1ds+βΓ0ϱ|v(x)|mrm1ds,(14) where α±=1,ifB>00,ifB0. Let ϑ be the smallest positive eigenvalue of problem (EVP2). Then for almost all ϱ ∈ (0, d) Ωϱvvr|v|m2|r=ϱdωΞ(m)ϱmϑ1mV±(ϱ),m>_2, where Ξ is defined by (12).

Proof

We repeat the proof of Lemma 2.3 [3] applying inequality (FW) instead of (W)m and taking β+ = α± 𝓑. □

Theorem 3.6

Let 𝓑 be defined by (5) and u be a weak solution of problem (QL), ϑ be the smallest positive eigenvalue of (EVP2). Let us assume that assumptions (a)-(d) are satisfied. In addition, assume that there exists real number kσ ≥ 0 defined by (13). Then there exist d ∈ (0,1) and a constant c > 0 independent of u such that for any ϱ ∈ (0, d) G0ϱ|u|qmm1|u|mdx+Γ0+ϱ1rm1|u|mm1(q+m1)ds+Γ0+ϱ1rm1|u|mm1(q+m1)dscΘm(ϱ), where Θ(ϱ)=ϱϑ~±,σ>ϑ~±ϱϑ~±ln1m(1ϱ),σ=ϑ~±ϱσ,σ<ϑ~±.(15)

Proof

The proof we starting from estimates the integrals on the right hand side of the inequality (5.8) of Theorem 4.2[3]. By the Young inequality with parametrs m,mm1, we get Γ0+ϱ|v||g(x,0)|ds1mδmΓ0+ϱ|v|mds+m1mδm1mΓ0+ϱ|g(x,0)|mm1dS,δ>0. Similarly, with δ = 1, we have Γ0ϱ|v||h(x,0)|ds1mΓ0ϱ|v|mds+m1mΓ0ϱ|h(x,0)|mm1ds and G0ϱvf(x)dx1mG0ϱ|v|mdx+m1mG0ϱ|f(x)|mm1dxϱmmϑV±(ϱ)+m1mG0ϱ|f(x)|mm1dx, in virtue of inequality (HW). Thus, by Lemma 3.5 and with regard to definition (14), from inequality (5.8) of Theorem 4.2[3], we have ςm1(1μς)bm1mω0m1G0ϱ|v|mdx+[β+b1+(m1)2m1m]Γ0+ϱ1rm1|v|mds+βΓ0ϱ1rm1|v|mdsςm1Ξ(m)mϑ1mϱV±(ϱ)+m1m(G0ϱ|f(x)|mm1dx+δm1mΓ0+ϱ|g(x,0)|mm1ds+Γ0ϱ|h(x,0)|mm1dS)+1m(ϱmϑV±(ϱ)+δmΓ0+ϱ|v|mds+Γ0ϱ|v|mds),δ>0.(16)

Now, there are two cases to consider: 𝓑 > 0 and 𝓑 ≤ 0. Firstly, let 𝓑 > 0. Therefore Γ0+ϱ|v|mds+Γ0ϱ|v|mdsϰϱm1V+(ϱ), where ϰ=max{1B;1B}. Hence, setting δ = 1, by assumption (i v), Lemma 3.5 with regard to definition (14) and (13), from (16) it follows that (K+δ(ϱ))V+(ϱ)Ξ(m)mϑ1mςm1ϱV+(ϱ)+m1mkσϱmσ,(17) where δ(ϱ) = const (m, q, ϑ, β, 𝓑)⋅ϱm−1 and 𝓚+ is defined by (11).

Next, let 𝓑 ≤ 0. Setting in (16) δ=[m2dm1β+b1+(m1)2m1m]1m, by assumption (d) and 1dm11rm1, we have ςm1(1μς)bm1mω0m1G0ϱ|v|mdx+12dm1[β;+b1+(m1)2m1m]Γ0+ϱ|v|mds+β;Γ0ϱ1rm1|v|mdsςm1Ξ(m)mϑ1mϱV(ϱ)+m1m(G0ϱ|f(x)|mm1dx+c1(m,b,β;+)Γ0+ϱ|g(x,0)|mm1ds+Γ0ϱ|h(x,0)|mm1ds)+1m(ϱmϑV(ϱ)+Γ0ϱ|v|mds).(18)

Finally, by Γ0ϱ|v|mdsϱm1V(ϱ), Lemma 3.5 and with regard to definition (14) and (13), from (18) it follows that (Kδ(ϱ))V(ϱ)Ξ(m)mϑ1mςm1ϱV(ϱ)+m1mkσϱmσ,(19) where δ(ϱ) = const(m, q, ϑ, β, 𝓑)⋅ϱm−1, and 𝓚 is defined by (10).

Thus, (17) and (19) we can write as follows (K±δ(ϱ))V±(ϱ)Ξ(m)mϑ1mςm1ϱV±(ϱ)+m1mkσϱmσ. The rest of the proof follows verbatim the proof of Theorem 4.2 [3], starting from the paragraph following inequality (5.14). □

3.4 The main result

Main result in this part of work is the following statement:

Theorem 3.7

Let u be a weak solution of problem (QL2) and assumptions (a)-(d) are fulfilled. Let us assume that M0=maxxG¯|u(x)| is known (see e.g. Theorem 3.1 [3]). In addition, suppose that there exist a real number kσ ≥ 0 defined by (13) and K ≥ 0 such that K=:supϱ>0ϱ2m1Θ(ϱ){ϱm(p2)p(m1)f(x)pm,G0ϱ1m1+ϱm1m2p(||g(x,0)||pm1,G0ϱ1m1+||h(x,0)||pm1,G0ϱ1m1)+ϱm2m+1m(m1)2p(||g(x,0)||pm1,G0ϱ1m1+||h(x,0)||pm1,G0ϱ1m1)}, where Θ(ϱ) is defined by (15) Then there exist d ∈ (0, 1) and a constant C > 0 independent of u such that |u(x)|C(|x|12mΘ(|x|))m1q+m1,xG0d.(20)

Proof

The estimate (20) we derive analogously to (6.1.7)[12] in virtue of Theorem 3.2 [3] and above proved Theorem 3.6 and the inequality (HW). □

4 Boundary value problems near the infinity

In [2] we consider the following boundary value problems for quasi-linear elliptic divergence equations: ddxiai(x,u,u)+b(x,u,u)=0,xGd;α(x)uv+1|x|m1γ(x|x|)u|u|q+m2=g(x,u),xGd;lim|x|u(x)=0,(QL3) where: d > 0, ai : Gd × ℝ × ℝn → ℝ, b : Gd × ℝ × ℝn → ℝ, m > 1, q ≥ 0, α(x)={0,ifxD;1,ifxD,DGd is the part of the boundary ∂Gd, where the Dirichlet boundary condition is posed, γ : (0,+∞) → (0,+∞) and uν=ai(x,u,u)cos(n,xi),n denotes the unit outward with respect to Gd normal to Gd.

Our aim was to find an exponent of (QL3) weak solutions’ decreasing rate at the infinity (in the case of the linear equation we refer to [13, 14]).

Definition 4.1

A function u is called a weak solution of problem (QL3) provided that uC0(Gd¯)Vm,01(Gd),lim|x|u(x)=0 and satisfies the integral identity Gd{ai(x,u,ux)ηxi+b(x,u,ux)η(x)}dx+Gdα(x)γ(ω)rm1u|u|q+m2η(x)ds=Gdα(x)g(x,u)η(x)ds for all functions ηC0(Gd¯)Vm,01(Gd),suchthatlim|x|η(x)=0.

4.1 Assumptions

Let m < n < p, 0μ<q+m1m1 be given numbers; a0(x), α1(x), b0(x) be non-negative measurable functions. We assume that:

  1. i=1nai2(x,u,ξ)+i=1n|ai(x,u,ξ)xi|2|u|q|ξ|m1+α1(x);α1(x)Lpm1(Gd)Lmm1(Gd);

  2. ai(x,u,ξ)ξi|u|q|ξ|ma0(x);a0(x)Lpm(Gd);

  3. |b(x,u,ξ)|μ|u|q1|ξ|m+b0(x);b(x,u,ξ)=β;(x,u)+b~(x,u,ξ),uβ;(x,u)|u|q+m;|b~(x,u,ξ)|μ|u|q1|ξ|m+b0(x);b0(x)Lpm(Gd)L1(Gd);

  4. g(x,u)u0;

  5. γ(ω) ≥ γ0 > 0 on ∂ Gd.

    In addition, suppose that the functions ai(x, u, ξ) are continuously differentiable with respect to u, ξ variables in Mϱ,M0=Gϱ¯×[M0,M0]×Rn,ϱ>R and satisfy in 𝔐ϱ, M0 the following conditions

  6. (m1)uai(x,u,ξ)u=qai(x,u,ξ)ξjξj;i=1,,n;

  7. i=1n|ai(x,u,ux)|u|q|u|m2uxi|2A(1|x|)|u|q|u|m1,

where 𝓐(t) is a monotonically increasing and Dini-continuous at zero function.

4.2 The main result

Suppose that there are finite numbers ks and K such that ks=supϱ>Rϱms{Gϱrq(m+1)(q+m)(m1)(a0(x))m(q+m1)(m1)(q+m)dx+Gϱrm+1m1(b0(x))mm1dx+Γϱα(x)(r|g(x,0)|)mm1ds},s>0;K=supϱ>Rϱnm1Θ(ϱ){ϱm(1np)q+m1(m1)(q+m)a0(x)pm,Gϱ2ϱq+m1(m1)(q+m)+ϱ(1np)mm1b0(x)pm,Gϱ2ϱ1m1+ϱ1npα1(x)pm1,Gϱ2ϱ1m1+ϱg(x,0),Γϱ2ϱ1m1}, where Θ(ϱ)=ϱϑ1m(m)Ξ(m)q+(m1)(1μ)q+m1,s>ϑ1m(m)Ξ(m)q+(m1)(1μ)q+m1;ϱϑ1m(m)Ξ(m)q+(m1)(1μ)q+m1ln1mϱ,s=ϑ1m(m)Ξ(m)q+(m1)(1μ)q+m1;ϱs,0<s<ϑ1m(m)Ξ(m)q+(m1)(1μ)q+m1,Ξ(m)=mm2(2m+2)m+22m,m2;(m1)m1m22m2,1<m2 and ϑ is the smallest positive eigenvalue of the eigenvalue problem for the m-Laplace-Beltrami operator on the unit sphere: divω(|ωψ|m2ωψ)+ϑ(m)|ψ|m2ψ=0,ωΩ;α(ω)|ωψ|m2ψν+γ(ω)|ψ|m2ψ(ω)=0,ωΩ.

Theorem 4.2

([2]). Let u be a weak solution ofproblem (QL3) and assumptions (1)(7) are satisfied. Suppose, in addition, that g(x,0)Ljj1(Gd),1<j<n1m1,d>0. Then there exist R~>R1 and a constant C0 > 0 such that for all xGR~ |u(x)|C0(|x|1nmΘ(|x|))m1q+m1.

5 The ideas of proofs

The ideas of proofs of Theorem 2.2, Theorem 3.7 and Theorem 4.2 are based on the deduction of new inequalities of Friedrichs-Wirtinger type with exact constants as well as some integral-differential inequalities adapted to our problems. The precise exponents of the solution’s decrease rate depend on these exact constants. For details we refer to [13].

The existence of the smallest positive eigenvalue of problem (EVP) for n = 3 was proved in [4]. The ideas of proof of this theorem are based on the Legendre spherical harmonics (see [4]) and the Gegenbauer functions.

Acknowledgement

The authors would like to thank Prof. Mikhail Borsuk for suggesting the problems and numerous helpful discussions during these studies.

References

  • [1]

    Bodzioch M., Borsuk M., Behavior of strong solutions to the degenerate oblique derivative problem for elliptic quasi-linear equations in a neighborhood of a boundary conical point, Complex Variables and Elliptic Equations, 2015, 60, No. 4, 510 – 528 CrossrefGoogle Scholar

  • [2]

    Borsuk M., Wiśniewski D., Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains, Central European Journal of Mathematics, 2012, 10, No. 6, 2051 – 2072 Web of ScienceGoogle Scholar

  • [3]

    Borsuk M., Żyjewski K., Nonlocal Robin problem for elliptic quasi-linear second order equations, Advanced nonlinear studies, 2014, 14, 159 –182 Google Scholar

  • [4]

    Bodzioch M., Oblique derivative problem for linear second-order elliptic equations with the degeneration in a 3-dimensional bounded domain with the boundary conical point, Electronic Journal of Differential Equations, 2012, 2012, No. 228, 1 – 28 Google Scholar

  • [5]

    Bodzioch M., Borsuk M., On the degenerate oblique derivative problem for elliptic second-order equations in a domain with boundary conical point, Complex Variables and Elliptic Equations, 2014, 59, No. 3, 324 – 354 CrossrefGoogle Scholar

  • [6]

    Lieberman G. M., Pointwise estimate for oblique derivative problems in nonsmooth domains, Journal of Differential Equations, 2001, 173, 178–211 CrossrefGoogle Scholar

  • [7]

    Lieberman G. M., Second order parabolic diffeential equations, World Scientific, Singapore-New Jersey-London-Hong Kong, 1996. Google Scholar

  • [8]

    Lieberman G. M., Trudinger N. S., Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. of AMS, 1986, 295, 509–546 CrossrefGoogle Scholar

  • [9]

    Borsuk M., Kondratiev V., Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains, North-Holland Mathematical Library, Elsevier, Amsterdam, vol 69, 2006 Google Scholar

  • [10]

    Borsuk M., Żyjewski K., Nonlocal Robin problem for elliptic second order equations in a plane domain with a boundary corner point, Appl. Math., 2011,38, No. 4, 369 – 411 Google Scholar

  • [11]

    Żyjewski K., Nonlocal Robin problem in a plane domain with a boundary corner point, Annales Universitatis Paedagogicae Cracoviensis, Studia Mathematica, 2011, X, 5 – 34 Google Scholar

  • [12]

    Borsuk M., Transmission Problems for Elliptic second-Order Equations in Non-Smoooth Domains, Frontiers in Mathematics, 2010. Google Scholar

  • [13]

    Wiśniewski D., Boundary value problems for a second-order elliptic equation in unbounded domains, Ann. Univ. Paedag. Cracov. Studia Math., 2010, IX, 87–122 Google Scholar

  • [14]

    Wiśniewski D., The behaviour of weak solutions of boundary value problems for linear elliptic second-order equations in unbounded cone-like domains, Ann. Math. Sil., 2016, 30, 203–217 Google Scholar

About the article

Received: 2016-10-29

Accepted: 2017-02-20

Published Online: 2017-05-23


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0033.

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© 2017 Bodzioch et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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