1,720,986 research outputs found
Liouville results for m-Laplace equations in a half plane in R2
No full textWe consider weak positive solutions of the equation in the half-plane with zero Dirichlet boundary conditions. Assuming that the nonlinearity is locally Lipschitz continuous and for , we prove that any solution is monotone. Some Liouville type theorems follow
in the case of Lane-Emden-Fowler type equations. Assuming also that is globally bounded, our result implies that solutions are one-dimensional, and the level sets are flat
Qualitative properties of solutions of m-Laplace systems
No full textWe prove regularity results for the solutions of the equation -Delta(m)u = h(x), such as summability properties of the second derivatives and summability properties of 1/vertical bar Du vertical bar. Analogous results were recently proved by the authors for the equation -Delta(m)u = f (u). These results allow us to extend to the case of systems of m-Laplace equations, some results recently proved by the authors for the case of a single equation. More precisely we consider the problem {-Delta(m1)(u) = f (v) u > 0 in Omega, u = 0 on theta Omega {-Delta(m2)(v) = g(u) v > 0 in Omega, v = 0 on theta Omega and we prove regularity properties of the solutions as well as qualitative properties of the solutions. Moreover we get a geometric characterization of the critical sets Z(u) equivalent to {x is an element of Omega vertical bar Du(x) = 0} and Z(v) equivalent to {x is an element of Omega vertical bar Dv(x) = 0}. In particular we prove that in convex and symmetric domains we have Z(u) = {0} - Z(v), assuming that 0 is the center of symmetry
A strong comparison principle for the p-laplacian
No full textWe consider weak solutions of the differential inequality of p-Laplacian type -(p)u - f(u) <= - Delta(p)v - f(v) such that u <= v on a smooth bounded domain in RN and either u or v is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that u < v on the boundary of the domain we prove that u < v, and assuming that u equivalent to v equivalent to 0 on the boundary of the domain we prove u < v unless u equivalent to v. The novelty is that the nonlinearity f is allowed to change sign. In particular, the result holds for the model nonlinearity f(s) = s(q) - lambda s(p-1) with q > p - 1
Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-laplace equations
No full textWe consider the Dirichlet problem for positive solutions of the equation -Delta(m)(u) = f (u) in a bounded smooth domain Omega, with f positive and locally Lipschitz continuous. We prove a Harnack type inequality for the solutions of the linearized operator, a Harnack type comparison inequality for the solutions, and exploit them to prove a Strong Comparison Principle for solutions of the equation, as well as a Strong Maximum Principle for the solutions of the linearized operator. We then apply these results, together with monotonicity results recently obtained by the authors, to get regularity results for the solutions. In particular we prove that in convex and symmetric domains, the only point where the gradient of a solution u vanishes is the center of symmetry (i.e. Z equivalent to {x is an element of Omega vertical bar D(u)(x) = 0} = {0} assuming that 0 is the center of symmetry). This is crucial in the study of m-Laplace equations, since Z is exactly the set of points where the m-Laplace operator is degenerate elliptic. As a corollary u is an element of C-2(Omega \ {0})
Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations
No full textAbstractWe consider the Dirichlet problem for positive solutions of the equation −Δm(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak C1(Ω̄) 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
A weak comparison principle in tubular neighbourhoods of embedded manifolds
No full textWe study weak solutions to degenerate quasilinear elliptic equations, involving first order terms, in unbounded tubular domains. In particular we show that, under suitable hypotheses, the weak comparison principle holds if the domain is narrow enough
Spectral theory for linearized p-Laplace equations
No full textWe continue and completely set up the spectral theory initiated in Castorina et al. [D. Castorina, P. Esposito, B. Sciunzi, Degenerate elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations, 34 (2009), 279–306] for the linearized operator arising from Δ_p u+f(u)=0. We establish existence and variational characterization of all the eigenvalues, and by a weak Harnack inequality
we deduce Hölder continuity for the corresponding eigenfunctions, this regularity being sharp. The Morse index of a positive solution can be now defined in the classical way,
and we will illustrate some qualitative consequences one should expect to deduce from
such information. In particular, we show that zero Morse index (or more generally, nondegenerate)
solutions on the annulus are radial
On Coron's problem for the p-Laplacian
Full text linkWe prove that the critical problem for the p-Laplacian operator admits a nontrivial solution in annular shaped domains with sufficiently small inner hole. This extends Coron's result [4] to a class of quasilinear problems
Radial symmetry for a quasilinear elliptic equation with a critical Sobolev growth and Hardy potential
Full text linkWe consider weak positive solutions to the critical p-Laplace equation with Hardy
potential in RN
−Δpu − γ
|x|p up−1 = up∗−1
where 1 < p < N, 0 γ < N−p
p
p
and p∗ = Np
N−p .
The main result is to show that all the solutions in D1,p(RN ) are radial and radially
decreasing about the origin
p-MEMS equation on the ball
No full textWe investigate qualitative properties of the MEMS equation involving the p−Laplace operator, 1 < p 2, on a ball B in R^N, N \geq 2. We establish uniqueness results for semi-stable
solutions and stability (in a strict sense) of minimal solutions. In particular, along the minimal
branch we show monotonicity of the first eigenvalue for the corresponding linearized operator and
radial symmetry of the first eigenfunction
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