We obtained that for every λ>0 in the minimum problems Iλ and I∞λ,there exists α∈0,λ,such that both problems Iα and I∞α have nontrivial solutions.

In this paper, a concentration-compactness lemma for the problem of quasilinear elliptic equations is given, and the existence of nontrivial solutions is discussed by use of this lemma.

With the mountain pass lemma and the means of straightening the boundary,the existence of nontrivial solutions are obtained by verifing the functional J(u) corresponding to the equations satisfy the local(PS) conditions.

Existence of nontrivial solution for an elliptic equation;

Existence of nontrivial solutions for the p-Laplacian Problem on unbounded domain;

On the nontrivial solutions and dead core problem for the equation Δu=c︱Du(X)︱~(p-1);

The relationship between the existence of the non-trivial solutions and the length of normal materials for the one-dimensional Ginzburg-Landau models of superconductivity with S-N-S junctions;

In this paper, we discuss one kind of nonlinear Volterra integral equation with the convolution kernel, and give some results on the uniqueness and approximate method of the non-trivial solutions of this equation.

The non-trivial solutions of equation λ_1x_1~k+λ_2x_2~k+…+λ_nx_k~n=0 are discussed, and the upper bound of the non-trivial solutions, when k≥6, is given here.

The enumeration problem for connective graphs with one nontrivial block was discussed by the theory of combinatorics,and the new conclusion is also given.