Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus

Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δ_pu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rⁿ with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rⁿ (see [10], [3] and [11]).     

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”     

On the multiplicity of positive solutions for p-Laplace equations via Morse theory

Let us consider the quasilinear problem

     

Regularity for p-harmonic equations with right-hand side in Orlicz-Zygmund classes

We establish regularity results for solutions of some degenerate elliptic PDEs, with right-hand side in a suitable Orlicz-Zygmund class. The nonnegative function which measures the degree of degeneracy of the ellipticity bounds is assumed to be exponentially integrable. We find that the scale of improved regularity is logarithmic and we indicate its exact dependence on the degree of the degeneracy of the problem.     

Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

We study the existence and nonexistence of positive (super)solutions to the nonlinear p-Laplace equation

     

On very weak solutions of degenerate p-harmonic equations

We establish a regularity result for very weak solutions of some degenerate elliptic PDEs. The nonnegative function which measures the degree of degeneracy of ellipticity bounds is assumed to be exponentially integrable. We find that the scale of improved regularity is logarithmic.      

On symmetry of nonnegative solutions of elliptic equations

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H  . Moreover, we prove that if u?0$u?0$, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set.     

On the convergence of global and bounded solutions of some evolution equations

It is shown that a special case of the well-known Lojasiewicz gradient inequality is sufficient to give a unified background for many convergence results in gradient or gradient-like systems appearing previously in the Literature. Besides as an illustration we give a direct proof of convergence in the case of 1D wave equations by a suitable adaptation of Zelenyak’s method.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

Regularity of flat free boundaries in two-phase problems for the p-Laplace operator

In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.     

Sufficient structure conditions for uniqueness of viscosity solutions of semilinear and quasilinear equations

In this article, we introduce a new approach for proving Maximum Principle type results for viscosity solutions of second-order, fully nonlinear possibly degenerate elliptic equations. This approach leads, in particular, to a better understanding of the conditions on the equation which are necessary to obtain such results. It allows us to provide new comparison results for semilinear and quasilinear equations.     

Regularity for a class of strongly degenerate quasilinear operators

We prove a Harnack inequality and regularity for solutions of a quasilinear strongly degenerate elliptic equation. We assume the coefficients of the structure conditions to belong to suitable Stummel–Kato classes.     

Multiplicity of positive solutions for a singular and critical problem

In this paper, we study the following singular, critical elliptic problem :

     

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in RN and in a bounded domain Ω⊂RN, with N?3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.     

Boundary estimates for solutions to operators of p-Laplace type with lower order terms

In this paper we study the boundary behavior of solutions to equations of the form

∇⋅A(x,∇u)+B(x,∇u)=0,     

Regularity of invariant sets in semilinear damped wave equations

Under fairly general assumptions, we prove that every compact invariant subset I of the semiflow generated by the semilinear damped wave equation

     

Renormalized solutions for nonlinear elliptic equations with variable exponents and L data

We prove existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and L¹$L^1$ data. The functional setting involves Lebesgue–Sobolev space with variable exponents W^1,p(⋅)(Ω)$W^{1,p(\cdot )}\left(\Omega \right)$.     

Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients

This paper deals with the behavior of two-dimensional linear elliptic equations with unbounded (and possibly infinite) coefficients. We prove the uniform convergence of the solutions by truncating the coefficients and using a pointwise estimate of the solutions combined with a two-dimensional capacitary estimate. We give two applications of this result: the continuity of the solutions of two-dimensional linear elliptic equations by a constructive approach, and the density of the continuous functions in the domain of the Γ-limit of equicoercive diffusion energies in dimension two. We also build two counter-examples which show that the previous results cannot be extended to dimension three.