Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

Some remarks on singular solutions of nonlinear elliptic equations. I

We study strong maximum principles for singular solutions of nonlinear elliptic and degenerate elliptic equations of second order. An application is given on symmetry of positive solutions in a punctured ball using the method of moving planes. Dedicated to Felix Browder on his 80th birthday     

A nondegeneracy result for a nonlinear elliptic equation

Let Ω be a smooth bounded domain of with N ≥ 5. In this paper we prove, for ɛ > 0 small, the nondegeneracy of the solution of the problem

Vector valued minimizers of anisotropic functionals: fractional differentiability and estimate for the singular set

We prove “fractional” higher differentiability for the gradient of minimizers of anisotropic integral functionals, if the growth exponents are no too far apart. This allows us to give an estimate for the Hausdorff dimension of the singular set of the minimizers.     

Nondegeneracy of elliptic problems with periodic nonlinearities in the plane

A class of asymptotically quadratic functionals on Hilbert spaces, called degenerate, is considered and explored. Our results are applied to obtain a extension, in the planar case, of a result published by Solimini in ‘On the solvability of some elliptic partial differential equations with the linear part at resonance’, J. Math. Anal. Appl., 117 (1986), 138-152. Similar extensions had been previously studied in the literature only for domains with particular geometries.     

Multiplicity of solutions for a noncooperative elliptic system with critical Sobolev exponent

In this paper, we study a system of elliptic equations by applying the Limit Index Theory. Under some assumptions on nonlinear part, we can obtain the existence of multiple solutions for the equations. The research is supported by NNSF of China (10471024) and Fujian Provincial Natural Science Foundation of China (A0410015).     

Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations

In this paper, we consider the problem of the existence of non-negative weak solution u of

Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems

We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is with Dirichlet boundary condition, where γ is the normalized Gaussian function in . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure. Partially supported by GMAMPA - INDAM, Progetto “Proprietà analitico geometriche di soluzioni di equazioni ellittiche e paraboliche”.     

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

Perturbation from symmetry for indefinite semilinear elliptic equations

We prove the existence of an unbounded sequence of solutions for an elliptic equation of the form \({-\Delta u=\lambda u + a(x)g(u)+f(x), u\in H^1_0(\Omega)}\), where \({\lambda \in \mathbb{R}, g(\cdot)}\) is subcritical and superlinear at infinity, and a(x) changes sign in Ω; moreover, g( ? s) =  ? g(s) \({\forall s}\). The proof uses Rabinowitz’s perturbation method applied to a suitably truncated problem; subsequent energy and Morse index estimates allow us to recover the original problem. We consider the case of \({\Omega\subset \mathbb{R}^N}\) bounded as well as \({\Omega=\mathbb{R}^N, \, N\geqslant 3}\).     

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form

An elliptic continuation result for harmonic functions in two dimensions

Let u be harmonic in a simply connected domainG ⊂ ℝ² and letK be a compact subset of G. In this note, it is proved there exists an “elliptic continuation” of u, namely there exist a smooth functionu ₁ and a second order uniformly elliptic operatorL with smooth coefficients in ℝ², satisfying:u ₁=u inK, Lu ₁=0 in ℝ². A similar continuation theorem, with u itself a solution to an elliptic second order equation inG, is also proved.     

On scalar field equations with critical nonlinearity

The paper concerns existence of solutions to the scalar field equation

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R ⁿ .     

Multiplicity theorems for superlinear elliptic problems

In this paper we study second order elliptic equations driven by the Laplacian and p-Laplacian differential operators and a nonlinearity which is (p-)superlinear (it satisfies the Ambrosetti–Rabinowitz condition). For the p-Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. Finally we indicate how in the semilinear case, Morse theory can be used to produce six nontrivial solutions. This paper was completed while the first author was visiting the University of Aveiro as an Invited Scientist. The hospitality and financial support of the host institution are gratefully acknowledged. The second and third authors acknowledge the partial financial support of the Portuguese Foundation for Science and Technology (FCT) under the research project POCI/MAT/55524/2004.     

Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions

Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂_ν u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L ^∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s| ^p ) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) =  s ^p then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of ∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential equation is of the form h(x, u) with h satisfying suitable growth conditions.     

Sign changing solutions of semilinear elliptic problems in exterior domains

We prove the existence of a sign changing solution to the semilinear elliptic problem , in an exterior domain Ω having finite symmetries.     

Regularity results for quasilinear elliptic equations in the Heisenberg group

We prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the Heisenberg group . The model case is the non-degenerate p-Laplacean operator where , and p is not too far from 2.     

Fast and slow decay solutions for supercritical elliptic problems in exterior domains

We consider the elliptic problem Δu  +  u ^p  =  0, u  >  0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded in , N ≥ 3, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay at infinity. In addition, a solution with fast decay O(|x|^2-N ) exists if p is close enough from above to the critical exponent.