We study the multiplicity of solutions to the elliptic equation Delta u+ f(x, u)= 0,under the assumption that f(x, u)/u crosses globally but not pointwise any eigenvalue for every x in a part of the domain ,when u varies from -infinity to infinity. Also we relax the conditions on uniform convergence of f(x, s)/s, which are essential in many results on multiplicity for asymptotically linear problems.

We define the topological degree for a class of operators and use it to find generalized zeros of operators. Using these results we get the existence of solutions for singular nonlinear elliptic equations.

Using a method developed by P´ erez and Wheeden and the representation of smooth functions by integral operators whose kernels are gradients of the Green functions, we obtain weighted Sobolev's inequalities for bounded domains which im- prove and unify several kinds of inequalities. From these results we establish Green functions and the existence, uniqueness and regularity results for a class of singular elliptic equations.

Our main result in this paper is the following: Given $H^m, H^n$ hyperbolic spaces of dimensional $m$ and $n$ corresponding, and given a Holder function $f=(s^1,...,f^{n-1}):\partial H^m\to \partial H^n$ between geometric boundaries of $H^m$ and $H^n$. Then for each $\epsilon >0$ there exists a harmonic map $u:H^m\to H^n$ which is continuous up to the boundary (in the sense of Euclidean) and $u|_{\partial H^m}=(f^1,...,f^{n-1},\epsilon)$.

We establish flows on normed spaces. Applying it we extend the results of Gromoll, Meyer, Morse and Palais for non-C-2 functionals.

We obtain the nonlinear versions of the Stampacchia theorem and the Lax–Milgram theorem. Our results are stronger than the classical ones even in the linear case. Applying these theorems we get nontrivial solutions of p-Laplace elliptic and pseudo-p-Laplace problems.

Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the C-1-smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.

We prove a version of Lagrange multipliers theorem for nonsmooth functionals defined on normed spaces. Applying these results, we extend some results about saddle point optimality criteria in mathematical programming.

This paper deals with the existence of a generalized solution in W01,p(Ω) to a nonuniformly nonlinear elliptic equation of the form -div(a(x,∇u))=f(x,u) in a bounded domain Ω of Rn. Here a satisfies |a(x,ξ)|⩽c0[h0(x)+h1(x)|ξ|p-1] for any ξ in Rn, a.e. x∈Ω, h0∈Lpp-1(Ω), and h1∈Lloc1(Ω).

We prove a discrete implicit mapping theorem and apply it to extend the results in (1) to the case of vector constraint functions. Our results can be applied to functions which are not C 1 -Frechet difierentiable neither Lipschitz continuous, even they are not continuous. Applying these results, we extend some results in (2).

We prove a Lagrange multipliers theorem for a class of functions that are derivable along directions in a linear subspace of a Banach space where they are defined. Our result is available for topological linear vector spaces and is stronger than the classical one even for two-dimensional spaces, because we only require the differentiablity of functions at critical points. Applying these results we generalize the Lax-Milgram theorem. Some applications in variational inequalities and quasilinear elliptic equations are given.

We extend some results of Amann about the topological degree for compact potential vector fields to potential operators of class (S)(+). Using these results, we get the existence of a third buckled state of a thin elastic shell in R-3 even in case the gradient of the potential energy of the shell might not be compact.

We define the topological degree for a class of operators and use it to solve a strongly singular nonlinear elliptic equation.

Searching minimizers of functions on the convenient level set of the constraint function we obtain generalized solutions of a non-homogeneous p-Laplace equation in border case without using the regularity results of linear elliptic equations.

We get the existence and regularity of minimizers of certain functionals, which may be degenerate and have non-polynomial growth. Applying these results we can find exponentially harmonic maps in every homotopy class of maps from a connected Riemannian C4-manifold into [Formula: see text], where [Formula: see text] is a compact Riemannian C4-manifold.

We study a class of quasilinear elliptic equations on the unit ball of ℝ n in the divergence form ∑ j=1 n D j{G(|x|2,|Du|2)D j u} =H(|x|) and get estimates on the boundary by using a modified barrier-function technique of Bernstein. We establish a maximum principle for the gradients of solutions and get a global gradient estimate. We prove that solutions with constant boundary condition must be radial. Finally, we apply these results to graphs {(x,u(x)):x∈H n } whereu:H n →ℝ is a smooth map of then-hyperbolic spaceH n =B(0,1) with the metric\(g = \frac{{4dx^2 }}{{(1 - \left| x \right|)^2 )^2 }}\) to get the existence of graphs with radial prescribed mean curvature.