On the Existence of Radially Symmetric Solutions for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ p $\end{document}-Laplace Equation with Strong Gradient Nonlinearities

We consider the Dirichlet problem for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ p $\end{document}-Laplace equation in presence of a gradient not satisfying the Bernstein–Nagumo type condition. We define some class of gradient nonlinearities, for which we prove the existence of a radially symmetric solution with a Hölder continuous derivative.


Introduction and the Main Results
Consider the Dirichlet problem for the p-Laplace equation where B R is a ball of radius R and ∂B R is the boundary of B R .Regarding the history of the issue and the first results on the existence and regularity of solutions to equations with p-Laplacian, we can refer to the monograph [1] by Ladyzhenskaya and Uraltseva.Variational, approximation, and topological methods are of use in studying boundary value problems for (1.1).Involving the method of the calculus of variations is due to the variational structure of the main part of (1.1).However, the presence of gradient terms in the equation essentially complicates the use of these methods.In this case, to prove the solvability of the boundary value problem of wide use are the topological methods based on a priori estimates, as well as various approximation methods.
In this article we will specifically focus on the dependence of F (x, u, ∇u) on the gradient of u.In this regard, we note the following works in which the study of boundary value problems was carried out in presence of gradient terms.
In [2][3][4][5], the existence of weak solutions to problem (1.1), (1.2) is proved on using approximation methods.In the works [6][7][8][9][10] similar results were proven on using the various topological methods based on theorems of Liouville type, the method of sub/supersolutions, and application of Krasnoselsky's Theorem.In [11], the results on the existence of solutions were obtained on using an iterative method that bases on the mountain pass theorem.In [12], the connection is studied between viscosity solutions and Sobolev solutions in equations with low order terms containing derivatives of the solution.The authors of [13,14] apply the Leray-Schauder Fixed Point Theorem to obtaining the existence of solutions by linearization methods, a priori estimates with weights, and Comparison Theorems.
Note that in all of the above works, the low order terms in the equation satisfy the Bernstein-Nagumo condition which in the case of equation (1.1) takes the form 3) with some constant c, provided that the solution satisfies max |u| ≤ M with some constant M .The novelty of our results lies in the proof of the existence of a solution in the case that the Bernstein-Nagumo condition is violated.Note however that, unlike the works mentioned above, we confine our research to radially symmetric solutions.
In [15,16] the existence of radially symmetric solutions to problem (1.1), (1.2) was proven when condition (1.3) is violated.Sufficient conditions for solvability in the specified class of functions were given, connecting the behavior of the nonlinear source and the convective term.
The novelty of the results of this article in comparison with [15,16] lies in the proof of the existence of a solution in the absence of the Bernstein-Nagumo condition for a significantly wider class of gradient nonlinearities.
We are interested in the existence of bounded radially symmetric solutions to problem (1.1), (1.2).We will assume that F (x, u, ∇u) can be represented in the form F (r, u, u r ) with the change of variables r = |x|.Some examples of these functions are as follows: In the sequel, we will simply denote the derivative of a function u with respect to r as u .It is well known that the radially symmetric solution of (1.1) satisfies the equation and boundary conditions Owing to the degeneracy (singularity) of equation (1.4), its solutions can fail to belong to the space of twice continuously differentiable functions.In this regard, let us define what we will mean by a solution of problem (1.4), (1.5).
Definition 1.1.Say that u(r) is a weak solution of problem (1.4), (1.5), if u (r) is Hölder continuous on [0, R], satisfies (1.5) and the integral identity Owing to the smoothness of the so-defined solution, we will understood (1.5) in the usual sense.
Let us make a few more comments related directly to the goal of this article.These remarks concern F (r, u, u ).We will construct a weak solution in the sense of Definition 1.1 by passing to the limit in a sequence of classical solutions of the corresponding regularized problem.The proof of the existence of classical solutions is carried out on the basis of a priori estimates and the Fixed Point Theorem.Obtaining a priori estimates begins with an a priori estimate of max |u|.The difficulty in obtaining such estimate arises in the case when the maximum principle cannot be directly applied to problem (1.4), (1.5).Obtaining an a priori estimate of max |u| for a regularized problem can be found in [15,16].To simplify the presentation and focus attention on the result of this work, which constitutes its novelty, we will assume that F has the form F (r, u, u ) = −u + g(r, u, u ) + f (r), g(r, u, 0) = 0. (1.6) In [16] we assume that the gradient term g(r, u, u ) is a continuous function satisfying the conditions g(s, u 1 , q) − g(r, u 2 , q) ≤ 0, q > 0, (1.7) for r > s and u 1 − u 2 > 0. In this article we will show that, owing to Lipschitz continuity in the spatial variable and strict monotonicity in u, it is possible in fact to remove the structural constraints on the gradient term associated with the behavior in the spatial variable to obtain the existence of a solution to problem (1.4), (1.5).
To formulate the main result we need some notations.Put f 0 = max r∈[0,R] |f (r)|, and let M be a constant satisfying the inequality (1.9) For the boundary estimate of the gradient of a solution we need the condition on g(r, u, u ) which follows from Lemma 1 of [16].Assume that g(r, u, u ) satisfies Let us introduce some more notations ) (1.13) Replace conditions (1.7) and (1.8) with where γ(r, u 1 , u 2 , q) ≥ γ 0 > 0. Introduce V as follows: for C sufficiently large, where C 0 is a positive constant.
Remark.In [16], there are given some conditions similar to (1.14) and (1.15) in the case that K and γ are constants.In this article we abandon the constancy of K and γ and add (1.16).Theorem 1.2.Assume that F (r, u, u ) is jointly continuous and satisfies (1.6).Under conditions (1.9)-(1.16),there is a weak solution of (1.4), (1.5) Remark.If (1.10) and (1.11) take place for an arbitrary M satisfying (1.9), then we can choose M in Theorem 1.2 to be equal to Let us give several simple examples of the problems of the form (1.4), (1.5) for which Theorem 1.2 holds.In all examples below, g(r, u, u ) does not satisfy (1.7) and (1.8).
The radially symmetric solution of (1.19) satisfies the equation (1.21) The radially symmetric solution of (1.21) satisfies the equation In order to prove Theorem 1.2, we regularize equation (1.4) and prove the classical solvability of the regularized problem, on using the technique of [17] and the fixed point principle.Next, we pass to the limit to obtain a weak solution of problem (1.4), (1.5).The article is organized as follows.
In Section 2 we obtain an a priori estimate of the classical solution of the regularized problem and its derivative.Section 3 contains a proof of the existence of a classical solution to the regularized problem (see Theorem 3.2), as well as the existence of a weak solution to problem (1.4), (1.5) in the sense of Definition 1.1 (see Theorem 1.2).

A Priori Estimates of a Solution of the Regularized Problem and Its Derivative
Instead of equation (1.4), we will consider its regularization where instead of F we will write its representation (1.6) The constant α ∈ (0, 1) is such that (u α ) As α we may take α = m k , where m is even and k is a positive integer.Moreover, we will assume that α > p − 1 for 1 < p < 2. Let us rewrite (2.1) in nondivergent form where Obviously a ε (z) = a ε (−z).We will study the existence of a classical solution to problem (2.1), (2.2).To this end, let us give a definition of this notion.
2) pointwise in (0, R) as well as the boundary conditions (1.5) understood in the usual sense.
Our goal in this section is to obtain some a priori estimates of a solution and its derivative that do not depend on the regularization parameter.The next lemma follows easily from the classical maximum principle.

Lemma 2.2. For every classical solution to problem (2.2), (1.5) we have the estimate
Note that such estimate of the maximum of a solution is not sufficient for obtaining an a priori estimate of its derivative near the boundary point r = R.Consider the function H(r) = M (R − r), where M is defined in (1.9).The following lemma is a special case of Lemma 1 of [16]: Lemma 2.3.Suppose that (1.9)-(1.11)take place.Assume that if 1 < p < 2 then ε < ε 0 = (1 + α − p) M α .Moreover, for each classical solution of (2.2), (1.5) we have Let us proceed with the estimate of the derivative of a classical solution of the regularized problem.Put where C is defined by (1.12), (1.13).It is easy to see that Lemma 2.4.Let the conditions of Lemma 2.3 be satisfied.Assume conditions (1.12)-(1.16).Then for every classical solution of (2.2), (1.5).