We give an expository review of recent results obtained for elliptic equations having natural growth terms of absorption type and singular data. As a new result, we provide an application to the case of lower order terms of subcritical growth, proving a general solvability result with measure data for a class of equations modeled on (1.6).
An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree...
We provide an explicit example of a nonlinear second order elliptic system of two equations in three dimension to compare two -regularity theories. We show that, for certain range of parameters, the theory developed in Daněček, Nonlinear Differential Equations Appl.9 (2002), gives a stronger result than the theory introduced in Koshelev, Lecture Notes in Mathematics,1614, 1995. In addition, there is a range of parameters where the first theory gives H"older continuity of solution for all , while...
We establish an existence theorem for a Dirichlet problem with homogeneous boundary conditions by using a general variational principle of Ricceri.
We prove an existence result for solutions of some class of nonlinear elliptic problems having natural growth terms and L¹ data.
We prove an existence result for some class of strongly nonlinear elliptic problems in the Musielak-Orlicz spaces , under the assumption that the conjugate function of φ satisfies the Δ₂-condition.
Let be a compact Riemannian manifold with boundary. We consider the problem (first studied by Escobar in 1992) of finding a conformal metric with constant scalar curvature in the interior and zero mean curvature on the boundary. Using a local test function construction, we are able to settle most cases left open by Escobar’s work. Moreover, we reduce the remaining cases to the positive mass theorem.
A recent multiplicity result by Ricceri, stated for equations in Hilbert spaces, is extended to a wider class of Banach spaces. Applications to nonlinear boundary value problems involving the p-Laplacian are presented.
We prove an integral estimate for weak solutions to some quasilinear elliptic systems; such an estimate provides us with the following regularity result: weak solutions are bounded.
We consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.