Positive stationary solutions of equations with p-laplace operator joint paper with Mateusz MACIEJEWSKI Nicolaus Copernicus University, Toruń, Poland Geometry in Dynamics (6th European Congress of Mathematics), July 1, 2012
Outline 1. Problem 2. Abstract setting 3. Topological degree 4. Index formulae 5. Existence results
1. Problem Consider div( u(x) p 2 u(x)) = f (x, u(x)), x Ω, u(x) 0, x Ω, u(x) = 0, x Ω where p 2, Ω R N (N 1) is a bounded domain with smooth Ω and f : Ω [0, + ) R is a Carathéodory function satisfying (f1) there is c > 0 with f (x, s) c(1 + s p 1 ) for all s 0, a.e. x Ω, (f2) f (x, 0) 0 for a.e. x Ω Remark: (1) We get rid of the usual assumption f (x, s) 0 for all s 0. (2) No monotonicity assumption involving f.
2. Abstract setting The PDE problem can be stated as an abstract one { Au = F(u), u M where A : X D(A) X is an operator on X := L p (Ω) given by Au := div( u p 2 u), u D(A), D(A) := {u W 1,p 0 (Ω) div( u p 2 u) exists and belongs to L p (Ω)}. M := {u X u(x) 0 for a.e. x Ω}. F : M X is given by F(u), v := f (x, u(x))v(x) dx, u M. Ω Define also N : X X by N(u), v := Ω u p 2 uv dx, v X.
2. Abstract setting Proposition (1) (A 1 ) N is homeomorphic, bounded on bounded sets and monotone (i.e. Au Av, u v 0 u, v X ); (A 2 ) A is maximal monotone and J α : X X, α > 0, J α (τ) := u, where u D(A) is the unique el. s.t. τ = (N+αA)(u), are well defined and continuous; J : X (0, + ) (τ, α) J α (τ) X is bounded on bounded sets and such that J X [α 1,α 2 ] is completely cont. if 0<α 1 α 2 ; (A 3 ) M := N(M) is an L-retract, i.e., r : B(M, η) M with η > 0 and a constant L > 0 such that r(τ) τ Ld M (τ) for all τ B(M, η); moreover J α (M ) M for all α > 0; (A 4 ) F is cont., bdd on bdd sets and satisfies the tangency cond. F(N 1 (τ)) T M (τ), for τ M.
2. Abstract setting The Bouligand tangent cone to M at τ M T M (τ) := α>0 α(m τ). Proposition (2) The tangency condition from (A 4 ) and the continuity of F N 1 imply that d M (ϱ + αf(n 1 (τ))) lim = 0 for all τ M. α 0 +, ϱ τ, ϱ M α Remark The solution of the problem Au = F(u) can be found as the fixed point N(u) + Au = N(u) + F(u) u = J 1 (N + F(u)). In the usual situation AleksanderifĆWISZEWSKI f 0, thenpositive F(M) stationary Msolutions, which of eq. enables with p-laplace usoperator
3. Topological degree However, in the general case when instead of f 0 we have (f2), the property F(M) M DOES NOT hold and the mapping J 1 (N + F) may take values out of M. In the abstract setting of conditions (A 1 ) (A 4 ), we construct a topological degree detecting solutions of { Au = F(u), u M D(A). We consider Φ α : M M, α>0, given by Φ α (u) := J α r(n(u) + αf(u)), u M.
3. Topological degree The tangency condition (see Prop. 2) and the L-retraction give the following Proposition (3) If U M is a bounded open set such that Au F(u) for all u M U D(A), then there exists α 0 > 0 such that for each α (0, α 0 ] Φ α (u) u for all u M U. Therefore we can put Deg M (A, F, U) := lim α 0 + Ind M(Φ α, U) where Ind M is the Granas fixed point index for mappings of M.
3. Topological degree Theorem (1) The defined number has the following properties: (i) (existence) if Deg M (A, F, U) 0, then there exists u U D(A) such that Au = F(u) (ii) (additivity) if U 1, U 2 are open disjoint subsets of a bdd open U M and 0 ( A + F)(U \ (U 1 U 2 ) D(A)), then Deg M (A, F, U) = Deg M (A, F, U 1 ) + Deg M (A, F, U 2 ); (iii) (homotopy invariance) if H : U [0, 1] X is a continuous and bounded mapping such that H(N 1 (τ), t) T M (τ) for all τ N(U), t > 0, and Au H(u, t) for all u M U D(A) and t [0, 1], then Deg M (A, H(0, ), U) = Deg M (A, H(1, ), U).
4. Index formulae (A) It is known that the eigenvalue problem { div( u p 2 u) = λ u p 2 u, x Ω, u(x) = 0, x Ω. does not admit any nonzero solutions if λ 0, i.e. the p-laplace has no nonpositive eigenvalues. (B) The smallest eigenvalue λ 1,p is given by the Rayleigh formula λ 1,p = inf u W 1,p 0 (Ω),u 0 Ω u(x) p dx Ω u(x) p dx. The eigenfunctions corresponding to λ 1,p are either strictly positive or negative in Ω and belong to L (Ω). (C) Moreover, λ 1,p is an isolated eigenvalue and if there are two eigenfunctions u, v for λ 1,p, then there exists α R such that u = αv. (D) If any eigenfunction does not change its sign in Ω, then the corresponding eigenvalue must be equal to λ 1,p.
4. Index formulae Theorem (2) If 2 p < and ρ L (Ω) is such that either ρ(x) > λ 1,p for a.e. x Ω, or ρ(x) < λ 1,p for a.e. x Ω, then Deg M (A, ρn, B M (0, R)) = { 1, if ρ(x)<λ1,p for a.e. x Ω, 0, if ρ(x)>λ 1,p for a.e. x Ω. The first equality, when ρ < λ 1,p, is straightforward (we join Φ α homotopically to a constant map). In order to prove the second, when rho > λ 1,p, one we need a nonexistence result. We shall use results due to Fleckinger et al. and the techniques of Lindqvist.
4. Index formulae By applying Lindqvist s techniques we show Lemma (1) Let v W 1,p 0 (Ω) be a nonnegative weak solution of { div( v p 2 v) = λ 1,p v p 2 u, x Ω, v(x) = 0, x Ω. and ρ L (Ω). If u W 1,p 0 (Ω) is a weak solution to then u L (Ω). div( u p 2 u) = ρ u p 2 u + v p 2 v on Ω,
4. Index formulae Fleckinger, Gossez, Takáč and de Thélin proved the following nonexistence result Lemma (2) If h L (Ω) is nonnegative and nonzero, then the equation div( u p 2 u) = λ 1,p u p 2 u + h, on Ω, has no nonzero weak solution in W 1,p 0 (Ω). We use this to obtain Lemma (3) If ρ L (Ω) and either ρ(x) > λ 1,p for a.e. x Ω or ρ(x) < λ 1,p for a.e. x Ω, then the problem div( u p 2 u) = ρ u p 2 u on Ω (1) does not admit a nonzero solution u W 1,p 0 (Ω) such that u 0.
4. Index formulae Theorem (2) and the homotopy invariance give the index formulae Theorem (3) Assume that there is ρ 0 L (Ω) such that f (x, s) lim s 0 + s p 1 = ρ 0 (x) uniformly with respect to x Ω. If either ρ 0 (x) < λ 1,p, for a.e. x Ω, or λ 1,p < ρ 0 (x), for a.e. x Ω, then there is δ > 0 s.t. A(u) F(u) for u D(A) (B M (0, δ)\{0}), and { 1, if ρ0 (x) < λ Deg M (A, F, B M (0, δ)) = 1,p for a.e. x Ω, 0, if ρ 0 (x) > λ 1,p for a.e. x Ω.
4. Index formulae Theorem (4) Assume that there is ρ L (Ω) such that f (x, s) lim s s p 1 = ρ (x) uniformly with respect to x Ω. If either ρ (x) < λ 1,p, for a.e. x Ω, or λ 1,p < ρ (x), for a.e. x Ω, then there is R > 0 s.t. A(u) F(u) for u D(A) (B M (0, R)\{0}) and { 1, if ρ (x) < λ Deg M (A, F, B M (0, R)) = 1,p for a.e. x Ω, 0, if ρ (x) > λ 1,p for a.e. x Ω.
5. Existence result The different indices at zero and infinity give the existence of nontrivial solution. Theorem (5) Suppose that a Carathéodory function f : Ω [0, + ) R and ρ 0, ρ L (Ω) satisfy (f1), (f2) and the following condition f (x, s) f (x, s) lim s 0 + s p 1 = ρ 0 (x) and lim s s p 1 = ρ (x) uniformly with respect to x Ω. If either ρ 0 (x) < λ 1,p < ρ (x), for a.a. x Ω, or ρ (x) < λ 1,p < ρ 0 (x), for a.a. x Ω, then the problem div( u(x) p 2 u(x)) = f (x, u(x)), x Ω, u(x) 0, x Ω, u(x) = 0, x Ω admits a nontrivial weak solution u W 1,p 0 (Ω).
5. Existence result Remarks (1) We get rid of the assumption f (x, s) 0 for all s 0, which is required by the Krasnosel skii fixed theorem (usually applied to problems of this kind). (2) No monotonicity assumption involving f (as in variational methods). (3) Our general setting works for problems without variational structure (for systems of equations).
Bibliography 1. Ćwiszewski A., Maciejewski M., Positive stationary solutions for p-laplacian problems with nonpositive perturbation, preprint http://arxiv.org/abs/1204.1295: 2. Ćwiszewski A., Kryszewski W., Constrained Topological Degree and Positive Solutions of Fully Nonlinear Boundary Value Problems, Journal of Differential Equations, Vol. 247 (8) (2009), 2235 2269. 3. Fleckinger J., Gossez J.-P., Takáč P., de Thélin F., Existence, nonexistence et principe de l antimaximum pour le p-laplacien, C. R. Acad. Sci. Paris Sér. I Math., 321 (1995), 731 734. 4. Lindqvist P., On the equation div( u p 2 u) + λ u p 2 u = 0, Proc. Amer. Math. Soc. 109 (1) (1990), 157 164. 5 Lindqvist P., ADDENDUM TO On the equation div( u p 2 u) + λ u p 2 u = 0, Proc. Amer. Math. Soc. 116 (2) (1992), 583 584.
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