We prove some weak and strong comparison theorems for solutions of differential inequalities involving a class of elliptic operators that includes the p-laplacian operator. We then use these theorems ...together with the method of moving planes and the sliding method to get symmetry and monotonicity properties of solutions to quasilinear elliptic equations in bounded domains.
Nous prouvons quelques théorèmes de comparaison faible et fort pour solutions de certaines inéqualités différentielles liées à une classe d’opérateurs elliptiques qui comprend le p-laplacien. Ces théorèmes sont utilisés avec la méthode de « déplacement d’hyperplanes å et la méthode de « translation å pour obtenir des propriétés de symétrie et de monotonie des solutions d’équations elliptiques quasilinéaires dans des domaines bornés.
We consider the Dirichlet problem for positive solutions of the equation −Δp(u)=f(u) in a convex, bounded, smooth domain Ω⊂RN, with f locally Lipschitz continuous.
We provide sufficient conditions ...guaranteeing L∞ a priori bounds for positive solutions of some elliptic equations involving the p-Laplacian and extend the class of known nonlinearities for which the solutions are L∞ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains.
We consider an elliptic problem of the type
$$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr ...{\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$ where Ω is a bounded Lipschitz domain in ℝN with a cylindrical symmetry, ν stands for the outer normal and $\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $. Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem. As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem
$$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$ For this one, we construct sequences of eigenvalues and provide variational characterization of them, following the usual approach for the Dirichlet case, but working in the product Hilbert space L2 (Ω) × L2(Γ2).
In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in an annulus in $\mathbb{R}^N$, $N \geq 2 $. More precisely we prove that ...solutions having Morse index $j \leq N $ are foliated Schwarz symmetric if the nonlinearity is convex and a full coupling condition is satisfied along the solution. PUBLICATION ABSTRACT
We consider weak nonnegative solutions to the critical p-Laplace equation in RN−Δpu=up⁎−1, in the singular case 1<p<2. We prove that if p⁎⩾2 then all the solutions in D1,p(RN) are radial (and ...radially decreasing) about some point.
In this paper we study the positive solutions of the equation −Δu + λu = f(u) in a bounded symmetric domain Ω in ℝN, with the boundary condition u = 0 on ∂Ω. Using the maximum principle we prove the ...symmetry of the solutions v of the linearized problem. From this we deduce several properties of v and u; in particular we show that if N = 2 there cannot exist two solutions which have the same maximum if f is also convex and that there exists only one solution if f(u) = up and λ = 0.
In the final section we consider the problem −Δu = uP + μuq in Ω with u = 0 on ∂Ω, and show that if 1 < p <N+2N-2, q ϵ0,1 there are exactly two positive solutions for μ sufficiently small and some particular domain Ω.
Dans ce travail nous étudions les solutions positives du problème -Δu+λ=f(u)dansΩu=0surlebordΩ où Ω est un domaine borné et symétrique dans ℝN. Avec l’aide du principe de maximum nous prouvons la symétrie des solutions v du problème linéarisé. A partir de ce résultat nous déduisons plusieurs propriétés de v et u; en particulier nous montrons que si f est convexe et N = 2 on ne peut pas avoir deux solutions différentes qui ont le même maximum. On prouve aussi qu’il y a une seule solution si f(u) = up et λ = 0.
Dans la dernière section nous étudions le problème -Δu=up+μuqdansΩu=0surleborddeΩ et montrons que si 1 < p < N + 2/N − 2, 0 < q < 1 et μ est petite il y a exactement deux solutions positives dans quelques domaines particuliers.
We consider the Dirichlet problem for positive solutions of the equation −Δ
m
(
u)=
f(
u) in a bounded smooth domain
Ω, with
f locally Lipschitz continuous, and prove some regularity results for weak
...C
1(
Ω
̄
)
solutions. In particular when
f(
s)>0 for
s>0 we prove summability properties of
1
|Du|
, and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |
Du|
m−2
. The point of view of considering |
Du|
m−2
as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov–Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions
u of the Dirichlet problem in bounded (and symmetric in one direction) domains when
f(
s)>0 for
s>0 and
m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<
m<2.
. In this paper we use the moving plane method to get the radial symmetry about a point (ProQuest: Formulae and/or non-USASCII text omitted; see image) of the positive ground state solutions of the ...equation (ProQuest: Formulae and/or non-USASCII text omitted; see image) in (ProQuest: Formulae and/or non-USASCII text omitted; see image) , in the case (ProQuest: Formulae and/or non-USASCII text omitted; see image) . We assume f to be locally Lipschitz continuous in (ProQuest: Formulae and/or non-USASCII text omitted; see image) and nonincreasing near zero but we do not require any hypothesis on the critical set of the solution. To apply the moving plane method we first prove a weak comparison theorem for solutions of differential inequalities in unbounded domains.PUBLICATION ABSTRACT
We consider sign changing solutions of the equation −Δm(u)=|u|p−1u in possibly unbounded domains or in RN. We prove Liouville type theorems for stable solutions or for solutions which are stable ...outside a compact set. The results hold true for m>2 and m−1<p<pc(N,m). Here pc(N,m) is a new critical exponent, which is infinity in low dimension and is always larger than the classical critical one.
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