We consider the existence of three non-trivial smooth solutions for nonlinear elliptic problems driven by the p-Laplacian. Using variational arguments, coupled with the method of upper and lower solutions, critical groups and suitable truncation techniques, we produce three non-trivial smooth solutions, two of which have constant sign. The hypotheses incorporate both coercive and non-coercive problems in our framework of analysis.

We establish the existence of positive solutions of the Sturm–Liouville problem

where

We assume g and to be non-negative, continuous functions, a(s) is a positive continuous function, c≥0, p>1, and the function h is sub-quadratic with respect to u′. We combine a priori estimates with a fixed-point result of Krasnosel'skii to obtain the existence of a positive solution.

Using variational methods, we obtain the existence of sign-changing solutions for a class of asymptotically linear Schrödinger equations with deepening potential well.

The existence of spherically symmetric solutions of the equation ▵u = f(u) is proved for a large class of functions f(z). Among others, functions satisfying an inequality zf(z) < 0 for |z| > A, and in particular the function f(z) = -sinh z, belong to this class.

We consider general boundary value problems for homogeneous elliptic partial differential operators with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. We also consider operators which are not properly elliptic and boundary systems which do not satisfy the complementing condition and determine when they give rise to left or right invertible operators. A priori inequalities and regularity results for the corresponding boundary value problems in Sobolev spaces are then readily obtained.

We study the existence, uniqueness and regularity of solutions to an exterior elliptic free boundary problem. The solutions model stationary solutions to nonlinear diffusion reaction problems, that is, they have compact support and satisfy both homogeneous Dirichlet and Neumann-type boundary conditions on the free boundary ∂{u > 0}. Then we prove convexity and symmetry properties of the free boundary and of the level sets {u > c} of the solutions. We also establish symmetry properties for the corresponding interior free boundary problem.

In this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

Let ui(x, t) be the ith component of a nonnegative solution of a weakly coupled system of second-order, linear, parabolic partial differential equations, for x ∈ Rn and 0 < t < T. We obtain lower estimates, near t = 0, for the Lebesgue measure of the set of x for which exceeds 1. Related results for Poisson integrals on a half-space are also described, some applications are given, and interesting comparisons emerge.

In this paper we give new results concerning the maximal regularity of the strict solution of an abstract second-order differential equation, with non-homogeneous boundary conditions of Dirichlet type, and set in an unbounded interval. The right-hand term of the equation is a Hölder continuous function.

We establish a method of constructing kernels of Bergman operators for second-order linear partial differential equations in two independent variables, and use the method for obtaining a new class of Bergman kernels, which we call modified class E kernels since they include certain class E kernals. They also include other kernels which are suitable for global representations of solutions (whereas Bergman operators generally yield only local representations).

We show that a warped product Mf = nf has higher rank and nonpositive curvature if and only if f is a convex solution of the Monge-Ampère equation. In this case we show that M contains a Euclidean factor.

The purpose of this note is to investigate traces of the functions In(1 +|u|), where u is a solution of a semi-linear partial differential equation of elliptic type, belonging to an appropriate Sobolev space. This article complements the results of Chabrowski and Thompson (1980), and Mihailov (1972), (1976).

The existence of positive solutions of some semilinear elliptic equations of the form −Δu = λf(u) is studied. The major results are a nonexistence theorem which gives a λ* = λ*(f,Ω) > 0 below which no positive solutions exist and a lower bound theorem for umax for Ω a ball. As a corollary of the nonexistence theorem that describes the dependence of the number of solutions on λ, two other nonexistence theorems, and an existence theorem are also proved.

In this paper we study the existence and uniqueness of positive solutions of boundary vlue problems for continuous semilinear perturbations, say f: [0, 1) × (0, ∞) → (0, ∞), of class of quasilinear operators which represent, for instance, the radial form of the Dirichlet problem on the unit ball of RN for the operators: p-Laplacian (1 < p < ∞) ad k-Hessian (1 ≤ k ≤ N). As a key feature, f (r, u) is possibly singular at r = 1 or u =0, Our approach exploits fixed point arguments and the Shooting Method.

We show that the Kato conjecture is true for m-accretive operators with highly singular coefficients. For operators of the form A = *F, where formally corresponds to d/dx + zδ on L2 (R), we prove that Dom (A1/2) = Dom() = e-zHH1(R) where H is the Heavysied function. By adapting recent methods of Auscher and Tchamitchian, we characterize Dom (A) in terms of an unconditional wavelet basis for L2(R).

We consider the interior and Dirichiet problems and problems with first order boundary conditions, for a second order homogeneous elliptic partial differential operator with constant coefficients. Under natural conditions on the operators, these problems give rise to isomorphisms between the appropriate spaces with homogeneous norms. From there we obtain a priori estimates and regularity results for boundary value problems in Sobolev spaces.

We prove two asymptotical estimates for minimizers of a Ginzburg-Landau functional of the form .

We prove a new energy or Caccioppoli type estimate for minimisers of the model functional ∫Ω|Du|2 + (det Du)2, where Ω ⊂ 2 and u: Ω → 2. We apply this to establish C∞ regularity for minimisers except on a closed set of measure zero. We also prove a maximum principle and use this to establish everywhere continuity of minimisers.

In this paper, first, we study the existence of the positive solutions of the nonlinear elliptic equations in unbounded domains. The existence is affected by the properties of the geometry and the topology of the domain. We assert that if there exists a (PS)c-sequence with c belonging to a suitable interval depending by the equation, then a ground state solution and a positive higher energy solution exist, too. Next, we study the upper half strip with a hole. In this case, the ground state solution does not exist, however there exists at least a positive higher energy solution.

The existence of positive solutions, vanishing at infinity, for the semilinear eigenvalue problem Lu = λ f(x, y) in RN is obtained, where L is a strictly elliptic operator. The function f is assumed to be of subcritical growth with respect to the variable u.