For a class of quasilinear Schrodinger equations we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method.

Solutions for Quasilinear Schrödinger Equations via the Nehari Method

We prove existence, regularity and nonexistence results for problems whose model is 

$$-\Delta u = \frac{f(x)}{u^{\gamma}}\quad {{\rm in}\,\Omega},$$

with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of ${\mathbb{R}^{N}}$. Here γ > 0 and f is a nonnegative function on Ω. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of γ (which can be equal, larger or smaller than 1).

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Semilinear elliptic equations with singular nonlinearities

We present a new approach to studying a class of quasilinear problems including the so-called Modified Nonlinear Schrodinger Equations (MNLS). We show that solutions of quasilinear equations can be obtained as limits of 4-Laplacian perturbations.

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Quasilinear elliptic equations via perturbation method

Introduction to Potential Theory. By L. L. Helms. Pp. ix, 282. £7. 1970. (Wiley.)

These lectures are devoted to a generalized critical point theory for nonsmooth functionals and to existence of multiple solutions for quasilinear elliptic equations. If f is a continuous function defined on a metric space, we define the weak slope |df|(u), an extended notion of norm of the Frechet derivative. Generalized notions of critical point and Palais-Smale condition are accordingly introduced. The Deformation Theorem and the Noncritical Interval Theorem are proved in this setting. The case in which f is invariant under the action of a compact Lie group is also considered. Mountain pass theorems for continuous functionals are proved. Estimates of the number of critical points of f by means of the relative category are provided. A partial extension of these techniques to lower semicontinuous functionals is outlined. The second part is mainly concerned with functionals of the Calculus of Variations depending quadratically on the gradient of the function. Such functionals are naturally continuous, but not locally Lipschitz continuous on H 0 1 . When f is even and suitable qualitative conditions are satisfied, we prove the existence of infinitely many solutions for the associated Euler equation. The regularity of such solutions is also studied.

Nonsmooth critical point theory and quasilinear elliptic equations

On p-convex sets and geodesics

We consider quasilinear elliptic equations involving the p-Laplacian and singular nonlinearities. We prove comparison principles and we deduce some uniqueness results.

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Existence and uniqueness for p-Laplace equations involving singular nonlinearities

We provide a variational approach to singular semilinear elliptic equations of the form †…u = u †‹ + g(x;u), for every ‹ > 0.

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A variational approach to a class of singular semilinear elliptic equations

We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian operator and singular nonlinearities.

Annamaria Canino

Papers

Nonsmooth critical point theory and quasilinear elliptic equations

On p-convex sets and geodesics

Existence and uniqueness for p-Laplace equations involving singular nonlinearities

A variational approach to a class of singular semilinear elliptic equations

Nonlocal problems with singular nonlinearity