Siegfried Carl

Sign-changing and multiple solutions for the

We obtain a positive solution, a negative solution, and a sign-changing solution for a class of p-Laplacian problems with jumping nonlinearities using variational and super-subsolution methods.

p

Abstract In this paper we consider an initial boundary value problem for a parabolic inclusion whose multivalued nonlinearity is characterized by Clarkes generalized gradient of some locally Lipschitz function, and whose elliptic operator may be a general quasilinear operator of Leray–Lions type. Recently, extremality results have been obtained in case that the governing multivalued term is of special structure such as, multifunctions given by the usual subdifferential of convex functions or subgradients of so-called dc-functions. The main goal of this paper is to prove the existence of extremal solutions within a sector of appropriately defined upper and lower solutions for quasilinear parabolic inclusions with general Clarkes gradient. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as tools from nonsmooth analysis.

-Laplacian

A variational approach to the method of upper and lower solution is suggested which allows to treat nonlinear elliptic boundary value problems with Baire-measurable lower order nonlinearities. To this end an associated multivalued setting of the problem is considered. First we prove the existence of solutions of a ‘truncated’ auxiliary problem which is related to the minimization of a nonsmooth functional whose critical points are shown to be solutions of this auxiliary problem. Then it is shown that any solution of the auxiliary problem solves the original one. The existence of critical points of the functional under consideration is proved by showing that it satisfies a generalized Palais-Smale condition which is suggested by the Variational Principle of Ekeland.

Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient

Abstract In this paper we consider an initial boundary value problem for a reaction–diffusion equation under nonlinear and nonlocal Robin type boundary condition. Assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic. Thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates, but yields also a measure of the convergence rate.

The weak upper and lower solution method for quasilinear elliptic equations with generalized subdifferentiable perturbations

Abstract This paper provides a new fixed point theorem for increasing self-mappings G : B → B of a closed ball B ⊂ X , where X is a Banach semilattice which is reflexive or has a weakly fully regular order cone X + . By means of this fixed point theorem, we are able to establish existence results of elliptic problems with lack of compactness.

Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions

Abstract We consider the Dirichlet boundary value problem for an elliptic inclusion governed by a quasilinear elliptic operator of Leray–Lions type and a multivalued term which is given by the difference of Clarkes generalized gradient of some locally Lipschitz function and the subdifferential of some convex function. Problems of this kind arise, e.g., in mechanical models described by nonconvex and nonsmooth energy functionals that result from nonmonotone, multivalued constitutive laws. Our main goal is to characterize the solution set of the problem under consideration. In particular we are going to prove that the solution set possesses extremal elements with respect to the underlying natural partial ordering of functions, and that the solution set is compact. The main tools used in the proofs are abstract results on pseudomonotone operators, truncation, and special test function techniques, Zorns lemma as well as tools from nonsmooth analysis.

Elliptic problems with lack of compactness via a new fixed point theorem

We consider discontinuous quasilinear elliptic systems with nonlinear boundary conditions of mixed Dirichlet-Robin type on the individual components. The system considered is of the general form Au + f(¢;u )= h, where A is a quasilinear elliptic operator of Leray-Lions type, u =( u 1;u2);and the vector fleld f =( f 1;f2 )i s assumed to be of mixed monotone type associated with competitive or cooperative species. The vector fleld f may be discontinuous with respect to all its arguments. The main goal is to prove the existence of solutions within the so-called trapping region. Furthermore, if, in addition, the components fk are continuous in their ofidiagonal (nonprincipal) arguments, one can show the compactness of the solution set within the trapping region. The main tools used in the proof of our main result are variational inequalities, truncation and comparison techniques employing special test functions, and Tarski’s flxed point theorem on complete lattices. Two applications of the theory developed in this paper are provided. The flrst application deals with the steady-state transport of two species of opposite charge within a physical channel, and in the second application a ∞uid medium is considered which may undergo a change of phase, and which acts as a carrier for certain solute species.

Quasilinear elliptic inclusions of hemivariational type: Extremality and compactness of the solution set

By combining the method of upper and lower solution and monotone iteration with variational in¬equality techniques a constructive existence result for elliptic BVPs with discontinuous nonlinearity is proved. The discontinuous nonlinearity f: R → R is supposed to admit a decomposition of the form f = g − h with functions g,h: R → R that are non-decreasing and in general discontinuous as well. Thus we are able to deal with nonlinearities f that may have jumps in both directions upward and downward which generalizes recent results e.g. of Ambrosetti and Turner [2], Ambrosetti and Badiale [3], Heikkila [17] and the author [8]

Trapping region for discontinuous quasilinear elliptic systems of mixed monotone type

Abstract First, we prove existence and comparison results for multi-valued elliptic variational inequalities involving Clarke’s generalized gradient of some locally Lipschitz functions as multi-valued term. Only by applying the definition of Clarke’s gradient it is well known that any solution of such a multi-valued elliptic variational inequality is also a solution of a corresponding variational-hemivariational inequality. The reverse is known to be true if the locally Lipschitz functions are regular in the sense of Clarke. Without imposing this kind of regularity the equivalence of the two problems under consideration is not clear at all. The main goal of this paper is to show that the equivalence still holds true without any additional regularity, which will fill a gap in the literature. Existence and comparison results for both multi-valued variational inequalities and variational-hemivariational inequalities are the main tools in the proof of the equivalence of these problems.

A combined variational-monotone iterative method for elliptic boundary value problems with discontinuous nonlinearity

Abstract We prove existence and comparison results for quasilinear parabolic inclusions with Clarke’s generalized gradient via appropriately defined sub-supersolution only assuming a local Lq-boundedness condition on Clarke’s gradient. The developed comparison principle allows us to show the equivalence of the considered parabolic inclusion with an associated evolutionary hemivariational inequality.