Saddle-shaped solutions of bistable diffusion equations in all of ℝ2m
Schrödinger-Poisson equations with supercritical growth.
Second-order boundary estimates for solutions to singular elliptic equations in borderline cases.
Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms
We are mainly concerned with equations of the form -Lu = f(x,u) + μ, where L is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, f satisfies the monotonicity condition and mild integrability conditions, and μ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric...
Semilinear elliptic problems in unbounded domains
We investigate the existence of positive solutions and their continuous dependence on functional parameters for a semilinear Dirichlet problem. We discuss the case when the domain is unbounded and the nonlinearity is smooth and convex on a certain interval only.
Singularly perturbed elliptic equations with solutions concentrating on a 1-dimensional orbit
We consider a singularly perturbed elliptic equation with superlinear nonlinearity on an annulus in , and look for solutions which are invariant under a fixed point free 1-parameter group action. We show that this problem can be reduced to a nonhomogeneous equation on a related annulus in dimension 3. The ground state solutions of this equation are single peak solutions which concentrate near the inner boundary. Transforming back, these solutions produce a family of solutions which concentrate...
Sobolev solution for semilinear PDE with obstacle under monotonicity condition.
Solutions of singular semilinear elliptic equations with critical weighted Hardy-Sobolev exponents
Some solutions are obtained for a class of singular semilinear elliptic equations with critical weighted Hardy-Sobolev exponents by variational methods and some analysis techniques.
Some continuation properties via minimax arguments.
Stability of the Pohožaev obstrucion in dimension 3
We investigate problems connected to the stability of the well-known Pohoˇzaev obstruction. We generalize results which were obtained in the minimizing setting by Brezis and Nirenberg [2] and more recently in the radial situation by Brezis and Willem [3].
Stable solutions of in
Several Liouville-type theorems are presented for stable solutions of the equation in , where is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.
Subcritical perturbations of resonant linear problems with sign-changing potential.
Symmetry of local minimizers for the three-dimensional Ginzburg–Landau functional
We classify nonconstant entire local minimizers of the standard Ginzburg–Landau functional for maps in satisfying a natural energy bound. Up to translations and rotations,such solutions of the Ginzburg–Landau system are given by an explicit solution equivariant under the action of the orthogonal group.