Elliptic Problems with Nonhomogeneous Differential Operators and Multiple Solutions
The present survey aims to report on recent advances in the study of non-linear elliptic problems whose differential part is expressed by a general operator in divergence form. The pattern of such differential operator is the p-Laplacian Δ p with \(1<p<+\infty\). More general operators can be considered, possibly having completely different properties, for instance not satisfying any homogeneity requirement. A major objective of our work is to provide existence theorems of multiple solutions for boundary value problems governed by such general operators. In this direction, a three nontrivial solutions theorem is presented. In the case of problems determined by the p-Laplacian, we give a theorem ensuring the existence of at least four nontrivial solutions. Moreover, a complete sign information is available: two positive solutions, a negative solution and a nodal (sign-changing) solution. Finally, we provide a theorem guaranteeing the existence of a positive solution for a problem involving the \((p,q)\)-Laplacian operator \(\Delta_p+\Delta_q\), with \(1<q<p\), and a nonlinearity depending on the solution and its gradient.
KeywordsNonlinear elliptic boundary value problem Nonhomogeneous differential operator p-Laplacian Eigenvalue problem Multiple solutions Critical point Minimizer Regularity Maximum principle
- 6.Chang, K.-C.: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 6. Birkhäuser, Boston, (1993)Google Scholar
- 16.Khan, A.A., Motreanu, D.: Local minimizers versus X-local minimizers. Optim. Lett. (2012). doi:10.1007/s11590-012-0474-8Google Scholar
- 19.Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, vol. 74. Springer-Verlag, New York (1989)Google Scholar
- 25.Motreanu, D., Winkert, P.: The Fu\v c\'ı k spectrum for the negative p-Laplacian with different boundary conditions. Nonlinear Analysis. Springer Optimization and Its Applications, vol. 68, pp. 471–485. Springer, New York (2012)Google Scholar