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

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.


Introduction
This is a brief description of results of [1][2][3] 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 ju.x/j Ä O.jxj˛/, near the boundary singularities.
Let G R n 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 and G d 0 is a rotational cone with the vertex at O and the aperture ! 0 2 .0; /, d 1, G R D fx D .r; !/ 2 R n j r 2 .R; 1/; ! 2 S n 1 ; n 2g; R 1; S n 1 is the unit sphere. We introduce the following notations for a domain G which has a boundary conical point: -WD G \ S n 1 , where S n 1 denotes the unit sphere in R n ; -@ : the boundary of ; -G b a WD G \ f.r; !/ W 0 Ä a < r < b; ! 2 g: a layer in R n ; -b a WD @G \ f.r; !/ W 0 Ä a < r < b; ! 2 @ g: the lateral surface of G b a ; -G d WD GnG d 0 ; d WD @Gn d 0 .
We use standard function spaces: C k .G/ with the norm juj k;G ; L p .G/ with the norm kuk p;G , p 1; the Sobolev space W k;p .G/ with the norm kuk W k;p .G/ for integer k 0; the weighted Sobolev-Kondratiev space V k p;˛. G/ for where the infimum is taken over all functions g such that Gˇ@ G D g in the sense of traces.

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 C 2 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: where E n denotes the unite exterior normal vector to @G R 0 nO, .r; !/ are spherical coordinates in R n with pole O; repeated indices are understood as summation from 1 to n.
, q > n and satisfies the equation for almost all x 2 G R " for all " > 0 as well as the boundary condition in the sense of traces on @G R 0 nO. We assume that M 0 D max ju.x/j is known (see e.g. Theorem 13.1 [7]).

The main result
where E is the unit exterior normal to @G 1 0 at the points of @ . Suppose that assumptions .A/ -.E/ are satisfied. Then there exist numbers d > 0, c 0 , c 1 independent of u such that - - and 4,5]). There exists the smallest positive eigenvalue of problem (EVP ), which satisfies the following inequalities for n 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 .EVP 2/) 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 of domain boundary @G is mapped by transformation on ./ and ./ \ @G ¤ ;: Let us consider the domain G R 0 R 2 : Moreover, let C and be the part of boundary @G R 0 for which x 2 > 0 and x 2 < 0 respectively. We assume, that @G R 0 D C [ is a smooth curve everywhere except at the origin O 2 @G R 0 and near the point O curves ˙are lateral sides of an angle with the measure ! 0 2 OE0; / and the vertex at O.
Furthermore, let be a diffeomorphism mapping of C onto † 0 : Additionally, we suppose that there exists d > 0 such that in the neighbourhood of d 0C the mapping is the rotation about the origin O and ! 0 2 is the angle of rotation.
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 C with its values of u on † 0 W here q 0; m > 1; 0; a 0 0;ˇ˙> 0; b 0 are given numbers and E n denotes the unit outward with respect to G R 0 normal to @G R 0 n O: Recall that we are dealing with the nonlocal problem and the boundary @G R 0 is non smooth, so the formulation of problem (QL 2 ) does not make sense in general. To make our formulation precise, we give the definition of the weak solution of (QL 2 ).

H W /
where # is the least positive eigenvalue of .EVP 2/ problem.

Assumptions
With regard to problem (QL 2 ) we assume that the following conditions are satisfied: (a) let p > m > 1; 0 Ä < qCm 1 m 1 be given numbers; g.x; u/ be the Caratheodory and continuously differentiable with respect to variable u function C R ! R; h.x; u/ be the Caratheodory and continuously differentiable with respect to variable u function R ! R; For the following we will use the numbers: and e #˙WD

Integral estimates
Proof. We repeat the proof of Lemma 2.3 [3] applying inequality .F W / instead of .W / m and takingˇC D˙B : where ı.%/ D const .m; q; #;ˇ ; B/ % m 1 ; and K is defined by (10). Thus, (17) and (19) we can write as follows The rest of the proof follows verbatim the proof of Theorem 4.2 [3], starting from the paragraph following inequality (5.14).

The main result
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 .H W /.

Boundary value problems near the infinity
In [2] we consider the following boundary value problems for quasi-linear elliptic divergence equations: is the part of the boundary @G d ; where the Dirichlet boundary condition is posed, W .0; C1/ ! .0; C1/ and @u @ D a i .x; u; ru/ cos.E n; x i /; E n denotes the unit outward with respect to G d normal to @G d : Our aim was to find an exponent of .QL 3 / weak solutions' decreasing rate at the infinity (in the case of the linear equation we refer to [13,14]). u.x/ D 0 and satisfies the integral identity x/ .!/ r m 1 ujuj qCm 2 Á.x/ds D Z @G d˛.

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 [1][2][3].
The existence of the smallest positive eigenvalue of problem (EVP ) for n D 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.