For semilinear elliptic equations of critical exponential growth we establish the existence of positive solutions to the Dirichlet problem on suitable non-contractible domains.
In the paper the differential inequality where , , is studied. Sufficient conditions on the function are established, which guarantee nonexistence of an eventually positive solution. The generalized Riccati transformation is the main tool.
We study the existence and nonexistence of positive solutions of nonlinear elliptic systems in an annulus with Dirichlet boundary conditions. In particular, a priori bounds are obtained. We also study a general multiple linear eigenvalue problem on a bounded domain.
We study the existence of positive solutions for the -Laplace Emden-Fowler equation. Let and be closed subgroups of the orthogonal group such that . We denote the orbit of through by , i.e., . We prove that if for all and the first eigenvalue of the -Laplacian is large enough, then no invariant least energy solution is invariant. Here an invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all invariant functions. Therefore...
In this paper we consider positive unbounded solutions of second order quasilinear ordinary differential equations. Our objective is to determine the asymptotic forms of unbounded solutions. An application to exterior Dirichlet problems is also given.
We consider linear elliptic equations in bounded Lipschitz domains with mixed boundary conditions on . The main feature of this boundary value problem is the appearance of both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient . We study positivity principles and anti-maximum principles. One of our main results states that if is somewhere negative, and then there exist two eigenvalues , such the positivity principle...