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A Dirichlet problem with asymptotically linear and changing sign nonlinearity.

Marcello Lucia, Paola Magrone, Huan-Song Zhou (2003)

Revista Matemática Complutense

This paper deals with the problem of finding positive solutions to the equation -∆[u] = g(x,u) on a bounded domain 'Omega' with Dirichlet boundary conditions. The function g can change sign and has asymptotically linear behaviour. The solutions are found using the Mountain Pass Theorem.

A new numerical model for propagation of tsunami waves

Karel Švadlenka (2007)


A new model for propagation of long waves including the coastal area is introduced. This model considers only the motion of the surface of the sea under the condition of preservation of mass and the sea floor is inserted into the model as an obstacle to the motion. Thus we obtain a constrained hyperbolic free-boundary problem which is then solved numerically by a minimizing method called the discrete Morse semi-flow. The results of the computation in 1D show the adequacy of the proposed model.

An elliptic equation with no monotonicity condition on the nonlinearity

Gregory S. Spradlin (2006)

ESAIM: Control, Optimisation and Calculus of Variations

An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree...

An Elliptic Neumann Problem with Subcritical Nonlinearity

Jan Chabrowski, Kyril Tintarev (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We establish the existence of a solution to the Neumann problem in the half-space with a subcritical nonlinearity on the boundary. Solutions are obtained through the constrained minimization or minimax. The existence of solutions depends on the shape of a boundary coefficient.

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Olivier Rey, Juncheng Wei (2005)

Journal of the European Mathematical Society

We show that the critical nonlinear elliptic Neumann problem Δ u μ u + u 7 / 3 = 0 in Ω , u > 0 in Ω , u ν = 0 on Ω , where Ω is a bounded and smooth domain in 5 , has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K , there exists μ K > 0 such that for 0 < μ < μ K , the above problem has a nontrivial solution which blows up at K interior points in Ω , as μ 0 . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

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