On montre que le faisceau des sursolutions locales dans d’un certain opérateur elliptique est maximal pour un principe du minimum adapté aux espaces de Sobolev. La continuité de la réduite variationnelle des éléments continus permet alors d’étudier des représentants s.c.i.
We investigate some nonlinear elliptic problems of the form where is a regular bounded domain in , , a positive function in , and the nonlinearity is indefinite. We prove the existence of solutions to the problem (P) when the function is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.
We prove the existence and uniqueness of weak solutions of boundary problem value problems in an unbounded domain Ω ⊂ Rn for strongly nonlinear 2m order elliptic differential equations.
The asymptotic behaviour is studied for minima of regular variational problems with Neumann boundary conditions on noncompact part of boundary.
We consider local minimizers of variational integrals like or its degenerate variant with exponents which do not fall completely in the category studied in Bildhauer M., Fuchs M., Calc. Var. 16 (2003), 177–186. We prove interior - respectively -regularity of under the condition that . For decomposable variational integrals of arbitrary order a similar result is established by the way extending the work Bildhauer M., Fuchs M., Ann. Acad. Sci. Fenn. Math. 31 (2006), 349–362.