L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document}-Norm Estimates of Weak Solutions via Their Morse Indices for the m-Laplacian Problems

This work is devoted to obtain the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} and the L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document}-estimates of solutions via their Morse indices to the following m-Laplacian problems -Δmu=f(x,u)inΩu=0,on∂Ω,(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _m u= f(x,u) \quad \text{ in }\quad \Omega \\ u=0, \quad \text{ on } \partial \Omega , \end{array}\right. }\qquad \qquad \qquad (1) \end{aligned}$$\end{document}where Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbf {R}}^N$$\end{document} is a bounded domain with smooth boundary, N>m>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N>m>2$$\end{document} and f∈C(Ω¯×R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C(\overline{\Omega }\times {\mathbb {R}})$$\end{document} which will be specified later. As far as we know, it seems to be the first time that such explicit estimates are obtained for a nonlinear degenerate problems. So, our main results extend and complement previously L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document}-estimates results in the literature.


Introduction
A priori L ∞ -estimate (also called a priori L ∞ -bound) is an estimate for the size of a solution. "A priori" is a Latin expression which means "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. A priori estimates play important role in the proof of existence of the solution. Indeed, when variational methods are not applicable, the existence of a solution can be proved using other methods as far as we can derive a priori L ∞ -bounds for all possible solutions (see [7,17]).
Ladyzhenskaya and Ural'tseva [14] proved that every weak solution u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) of (1) belongs to C β (Ω) for some β ∈ (0, 1), and a more minute estimate such that u ∈ C 1+α (Ω) for some α ∈ (0, 1) can be derived from the result of Tolksdorf [19]. Furthermore, DiBenedetto [3] and Lewis [13] assures that every bounded weak solution of (1) enjoys C 1+α loc regularity. There exist various methods which provide a priori L ∞ -estimates of solutions of elliptic problems: In [16],Ôtani obtained an L ∞ -estimate of weak solutions which the proof of this regularity result relies on Moser's iteration sheme and a certain "Pohozaev-type inequality" which is valid for all weak solutions in P , where P := {u ∈ W 1,p 0 (Ω) L q (Ω); x i |u| q−2 u ∈ L p p−1 (Ω), i = 1, 2, . . . , N} ⊂ W 1,p(Ω) 0 ∩ L ∞ (Ω). Another method is the Morse index which was first explored (in harmonic equations and the subcritical case) by Bahri and Lions [2] and extended by Farina [8] to 1 < p < p c (N ), where p c (N ) is the so-called Joseph-Lundgren exponent. The basic argument used in the previous works is blow-up technique which was first introduced by Gidas and Spruck [9]. Motivated by [2], based on some local interior estimates and careful boundary estimates, Yang obtained in a famous paper [20] the first explicite L ∞ -bounds via the Morse index and he controlled the L p and L ∞ norm of solution by polynomial growth in Morse index. Note that the explicite L ∞ -estimate was improved in [10] under a large assumptions and extended in [15] to the bi-harmonic and triharmonic semilinear problems and for the polyharmonic [11] for any classical solution u. Inspired by the previously mentioned works, we are concerned to establish some L p and L ∞ estimates for weak solution of (1) via their Morse indices. However, in this work, we use a different approach to those explored in [15,20] because here we are working with the m-Laplacian operator which is a nonlinear degenerate operator and a standard boot-strap iteration do not yield any extra regularity of the solution which only gives C 1,α loc (Ω) regularity, for some α ∈ (0, 1) (we refer the interested readers to [4][5][6]12,16,18]). Therefore, Eq. (1) is to be understood with its weak formulation.
Formally, The linearized operator of (2) at u is given by for all v, ϕ ∈ C 1 c (Ω). We observe that if we consider  (4) then ρ ∈ L 1 loc (Ω) by the C 1,α regularity of u. Observe that, this weight is 1 when m = 2, so we will be mostly concerned with the case m > 2, in the case 1 < m < 2 the weight |∇u| m−2 might not belong to the space L 1 loc (Ω). This prevents the use of classical estimates in weighted Sobolev spaces and in this case it is not even clear which definition of Morse index would be the natural one.
Before stating our results, we present first our assumptions on the nonlinearity f .
Let f ∈ C(Ω × R) and f (x, s) = ∂f ∂s (x, s) exists for all (x, s) ∈ Ω × R such that: (h 1 ) (Subcritical growth) there exists 0 < θ < 1 such that for |s| > s 0 and From (h 0 ) and (h 1 ), we may easily see that there exists two positive constants C 1 and C 2 such that: |f Our first result is to claim an explicite estimate of |∇u| L m (Ω) via the Morse index: The proof of Theorem 1.1 is based on local estimate from a combination of the quadratic form and (1) and a variant of the Pohozaev identity (see [4,16]) which is key point to finish the proof. Precisely, we will test the Eq. (1) against ψ∇u · x, where ψ is an appropriate cut-off function allowing to avoid the spherical integrals raised in Yang's proof [20] which are very difficult to control. We will adapt the Moser iteration technique [1,16] to prove that the L ∞ norm evolves less rapidly than polynomial growth of the Morse indices of solutions.

Notations and Preliminaries
Consider a cut-off function ψ ∈ C 1 c (B R (y)) with 0 ≤ ψ ≤ 1 where y ∈ R N and B R (y) = B(y, 3R 7 ) denotes the open ball of center y and radius 3R 7 . We denote ∂Ω R (y) := ∂Ω ∩ B R (y), v the unit normal vector on ∂Ω and for x ∈ Ω ∩ B R (y), To derive a global estimate from local controls on Ω ∩ B R (y), we will cover Ω\Ω 2 (R) with finite number of balls B R (y), y ∈ Ω 1 (R) which yields in particular that ∂Ω R (y) = ∅. However we will cover Ω 2 (R) by B R (y) with So we need first to control the following boundary term raised in the next variant of Pohozaev identity.

Proof of Theorem 1.1
Proof. First we establish the following L p -estimate via the Morse index which will be combined with Lemma 2.2 to prove Theorem 1.1. Let u ∈ C 1,α loc (Ω) be a weak solution of Eq. (1), R 0 > 0 obtained in Lemma 2.2 and y ∈ Γ(R 0 ) Ω 1 (R 0 ). We denote by A b a = {x ∈ R N ; a < |x − y| < b}. For j = 1, 2, . . . , i(u) + 1, we designate by Let ϕ j be a family of C 1 functions satisfying the following conditions: We will use the test function φ k  If f satisfies (h 0 ), then there exists a positive constant C = C(Ω, f) and j 0 ∈ {1, 2, · · · , 1 + i(u)} such that Proof. Let k > m will be defined later. It is easy to see that supp(uφ m j ) ∩ supp(uφ m l ) = ∅, ∀ 1 ≤ l = j ≤ 1 + i(u), so the family uφ m j , 1 ≤ j ≤ 1 + i(u) are mutually orthogonal for the quadratic form Λ u (see (3)) . Hence, by the definition of i(u) defined by (5), there exists j 0 ∈ {1, 2, . . . , 1 + i(u)} such that Applying Young's inequality and using (h 0 ) we derive that for any > 0 there is C ,m > 0 such that Multiplying now (1) by uφ 2m j0 and integrating by parts, we have As above using Young's inequality, there holds Multiplying (21) by (m − 1 + μ ) with 0 ≤ μ ≤ μ and combining with (20), we obtain Now substituting in the last inequality, φ j0 by φ k j0 (where k > m will be defined later), then (18) implies Applying Young's inequality, we derive that for any > 0 there is Choose k = m+μ 2μ so that 2mk = (m + μ)(2k − 1) and using (7), we deduce that For > 0 sufficiently small, we obtain Now we turn to the proof of Theorem 1.1. Set ρ := where a j0 and b j0 are defined in (17). So applying Lemmas 2.2 with ψ = ψ j0 and R = R 0 , we derive that In view of Young's inequality, (h 0 ) and (7), the above inequality implies Thus, according to (19), we obtain By covering argument (see (10)), we get finally Therefore, Theorem 1.1 follows.