Search results for "MINIMIZERS"
showing 6 items of 6 documents
Nonlinear concave-convex problems with indefinite weight
2021
We consider a parametric nonlinear Robin problem driven by the p-Laplacian and with a reaction having the competing effects of two terms. One is a parametric (Formula presented.) -sublinear term (concave nonlinearity) and the other is a (Formula presented.) -superlinear term (convex nonlinearity). We assume that the weight of the concave term is indefinite (that is, sign-changing). Using the Nehari method, we show that for all small values of the parameter (Formula presented.), the problem has at least two positive solutions and also we provide information about their regularity.
Symmetry of minimizers with a level surface parallel to the boundary
2015
We consider the functional $$I_\Omega(v) = \int_\Omega [f(|Dv|) - v] dx,$$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_\Omega$ admits a minimizer in $W_0^{1,1}(\Omega)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differenti…
Radial symmetry of p-harmonic minimizers
2017
"It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557--611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy". The quotation is from [J. Sivaloganathan and S. J. Spector, Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity, Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008), no. 1, 201--213] and seems to be still accurate. The model case of the $p$-harmonic energy is considered here. We prove that the planar radial minimizers are indee…
Multiple solutions for parametric double phase Dirichlet problems
2020
We consider a parametric double phase Dirichlet problem. Using variational tools together with suitable truncation and comparison techniques, we show that for all parametric values [Formula: see text] the problem has at least three nontrivial solutions, two of which have constant sign. Also, we identify the critical parameter [Formula: see text] precisely in terms of the spectrum of the [Formula: see text]-Laplacian.
The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces
2020
In the planar setting the Rad\'o-Kneser-Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Rad\'o-Kneser-Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expressio…
On a Robin (p,q)-equation with a logistic reaction
2019
We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter \(\lambda \gt 0\) varies. Also, we show that for every admissible parameter \(\lambda \gt 0\), the problem admits a smallest positive solution.