Existence de solutions à support compact pour un problème elliptique quasilinéaire

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1 Existence de solutions à support compact pour un problème elliptique quasilinéaire Jacques Giacomoni, Habib Mâagli, Paul Sauvy Université de Pau et des Pays de l Adour, L.M.A.P. Faculté de sciences de Tunis, Département de Mathématiques 18 ième colloque C.S.M.T.

2 Framework Let Ω a bounded domain of R N, N 2 with a C 2 boundary. We consider the following quasilinear elliptic problem: { r u = K(x)(λu (P λ ) p u q ), u 0 in Ω, u = 0 on Ω. r u := div ( u r 2 u ), r > 1 is the r-laplacian operator. λ > 0 is a real parameter. 1 < q < p < r 1. K : Ω R is a positive function having a singular behaviour near the the boudary Ω, more precisely: K(x) d(x, Ω) k, with k (0,r).

0 on Ω. r u := div ( u r 2 u ), r > 1 is the r-laplacian operator. λ > 0 is a real parameter.

3 Objectives We call u W 1,r 0 (Ω), a weak solution of (P λ) if and only if: ϕ D(Ω), u r 2 u. ϕ dx = K(x)(λu p u q )ϕ dx. Goal: Origins: Ω Ω Existence of positive or compact support solutions in function of the blow-up rate k (0,r) of K(x). Population dynamics. Chemical reactions. Plasma physics. Previous work: Yang Haitao in [1] for the Laplacian case (when r=2).

Goal: Origins: Ω Ω Existence of positive or compact support solutions in function of the

4 Theorem 1 When k < 1+q, there exists a constant Λ 1 > 0 such that: For λ > Λ 1, (P λ ) has a minimal positive solution u λ W 1,r 0 (Ω) C( Ω ) which is increasing with respect to λ. For λ < Λ 1, (P λ ) has no positive solution. Theorem 2 When q > r(k 2)+1 r 1 and k [1+q,r), there exists Λ 2 > 0 such that: For λ > Λ 2, (P λ ) has a compact support solution u λ W 1,r 0 (Ω) C1,α( Ω ), α > 0 which is increasing with respect to λ. For λ < Λ 2, (P λ ) has no non trivial solution.

Theorem 2 When q > r(k 2)+1 r 1 and k [1+q,r), there exists Λ 2 > 0 such that: For λ > Λ 2, (P λ ) has a compact

5 Preliminary results Non existence lemma: Lemma 3 For k (0,r), there exists λ > 0 such that (P λ ) has no non-trivial solution for λ < λ.

6 Preliminary results Construction of a supersolution for (P λ ): We consider the following problem: { r u = λk(x)u (Q λ ) p, u 0 in Ω, u = 0 on Ω. Lemma 4 If k ]0,1+p[, there exists a unique ū C 1,α( Ω) solution of (Q λ ) satisfying u ϕ 1 in Ω, for an α ]0,1[. If k = 1+p, there exists a unique u W 1,r 0 (Ω) C ( Ω ) ) 1 solution of (Q λ ) satisfying u ϕ 1 ln( A r k ϕ 1 in Ω, with A > 0 sufficiently large. If k ]1+p,r[, there exists a unique u W 1,r 0 (Ω) C ( Ω ) r k solution of (Q λ ) satisfying u ϕ r (1+p) 1 in Ω.

Lemma 4 If k ]0,1+p[, there exists a unique ū C 1,α( Ω) solution of (Q λ ) satisfying u ϕ 1 in Ω, for an α ]0,1[.

7 Proof of Theorem 1 The proof is based on a sub and supersolution method: By the previous lemma, u C 1,α( Ω ) is a supersolution of (P λ ), moreover u ϕ 1 in Ω. We prove that u = Mϕ 1 τ, with a τ > 1 is a subsolution of (P λ ) for M > 0 sufficientlly large. Problem: K(x)(λu p u q ) blows up when d(x) 0 and we can t directly apply a sub-supersolution method on Ω. Idea: Apply a sub-supersolution method at the interior of Ω.

We prove that u = Mϕ 1 τ, with a τ > 1 is a subsolution of (P λ ) for M > 0 sufficientlly large.

8 Proof of Theorem 1 Because of that, let us itroduce (Ω k ) k N Ω an increasing sequence of regular subdomains of Ω such that: Ω k Ω, that is to say sup d(x,ω) 0. k + x Ω k k + d( Ω, Ω k ) > 1 k, for k sufficiently large. Then by increasing iterative schemes on the Ω k, we prove that (P λ ) as a minimal positive solution u W 1,r 0 (Ω) C( Ω ) satisfying u u u a.e. in Ω.

k + x Ω k k + d( Ω, Ω k ) > 1 k, for k sufficiently large.

9 Proof of Theorem 2 Step 1: Existence of a non trivial solution of (P λ ) If we directly study the functionnal : I λ (v) = 1 v r dx+ 1 K(x) v q+1 dx λ r q +1 p +1 Ω Ω with v W 1,r 0 (Ω), we can t hope to obtain a bounded solution. Ω K(x) v p+1 dx, Idea: Use a Perron s method and define a new funtionnal cut from above.

+1 Ω Ω with v W 1,r 0 (Ω), we can t hope to obtain a bounded solution.

10 Proof of Theorem 2 Let ϕ 0 D(Ω), ϕ 0 0 such that for λ > 0 sufficiently large, I λ (ϕ 0 ) > 0 in Ω. Let M > 0 such that Mu ϕ 0 in Ω, with u W 1,r 0 (Ω) C( Ω ) solution of (Q λ ). Then we can define: K(x)[λ(Mu) p (Mu) q ] if Mu u, f λ (x,u) = K(x)[λu p u q ] if 0 u Mu, 0 if u 0. and F λ (x,v) = v 0 f λ (x,u) du.

11 Proof of Theorem 2 Then we consider the functionnal E λ (v) = 1 v r dx r Ω Ω F λ (x,v)dx, v W 1,r 0 (Ω). The condition q > r(k 2)+1 r 1 implies that E λ is well difined on W 1,r 0 (Ω). By Hölder s and Hardy s inequality, we prove that E λ is bounded from below. We finaly prove that there exists u W 1,r 0 (Ω) such that E λ (u ) = min E λ (v) and 0 u Mu a.e. in Ω. v W 1,r 0 (Ω) Then, E λ (u ) E λ (ϕ 0 ) = I λ (ϕ 0 ) < 0 and by standard calculus of variation method, u is a non-trivial solution of (P λ ).

By Hölder s and Hardy s inequality, we prove that E λ is bounded from below.

12 Proof of Theorem 2 Step 2: compact support of the solution u. We prove that there exists α > 1 and M > 0 such that u (x) Mϕ 1 (x) α a.e. for dist(x, Ω) sufficiently small. The reaction term λk(x)u p will not be large enough to involve the positivity of the solution near the boundary. See Alvarez, Díaz [2]. Since ε 0 F(s) 1 r ds < +, with F(s) = s 0 u q du, we can construct a supersolution w W 1,r (Ω) of (P λ ) with a compact support such that u w near the boundary. This argument also appears in Vásquez [3]. By a classic regularity result due to Lieberman [4], we finaly have u W 1,r 0 (Ω) C1,α( Ω ).

Since ε 0 F(s) 1 r ds < +, with F(s) = s 0 u q du, we can construct a supersolution w W 1,r (Ω) of (P λ ) with a compact support such that u w near the

13 For the elliptc problem (when r = 2), extend the previous results by considering weight K(x) in the class of Kato functions i.e. m ( K(x) d(x, Ω) (log k n n=1 A d(x, Ω) )) µn, with 0 < k < 2 and n {1,...,m}, µ n R, or k = 2, µ 1 = µ 2 = = µ l 1 = 1 and n {l,,m}, µ n R. log n = log log log (n times) and A > 0 sufficiently large. This work is based on recent results due to Gontara, Mâagli, Masmoudi and Turki [5]. Study of the regularity of the solutions. Same approach for the parabolic problem related to (P λ ).

This work is based on recent results due to Gontara, Mâagli, Masmoudi and Turki [5]. Study of the regularity of the solutions.

14 Haitao, Yang. Positive versus compact support solutions to a singular elliptic problem, J.Math. Appl. 319 (2006) Alvarez Luís, Jesús Ildefonso Díaz. On the behaviour near the free boundary of solutions of some nonhomogeneous elliptic problems, (1986) Vázquez, Juan Luis, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12, 1984, Lieberman, Gary M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12, (1988) Gontara Sabrine, Mâagli Habib, Masmoudi Syrine, Turki Sameh. Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem, J.Math. Appl (2010).

elliptic equations, Appl. Math. Optim., 12, 1984, 191-202. Lieberman, Gary M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable