We study the semilinear problem with the boundary reaction −∆u + u = 0 in Ω, ∂u ∂ν = λf(u) on ∂Ω, where Ω ⊂ R N , N > 2, is a smooth bounded domain, f : [0, ∞) → (0, ∞) is a smooth, strictly positive, convex, increasing function which is superlinear at ∞, and λ > 0 is a parameter. It is known that there exists an extremal parameter λ ∗ > 0 such that a classical minimal solution exists for λ < λ∗ , and there is no solution for λ > λ∗ . Moreover, there is a unique weak solution u ∗ corresponding to the parameter λ = λ ∗ . In this paper, we continue to study the spectral properties of u ∗ and show a phenomenon of continuum spectrum for the corresponding linearized eigenvalue problem.
In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $\mathbb{R}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $\mathbb{R}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally, in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).
The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub- and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub- and supersolutions are established.
In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that their solution set is a compact $R_{\delta }$-set in $(CT,L^2(Z))$.