Bifurcations of homoclinic orbit connecting the strong stable and strong unstable directions are investigated for four-dimensional system. The existence, numbers, co-existence and incoexistence of 1-homoclinic orbit, 2n-homoclinic orbit, 1-periodic orbit and 2n-periodic orbit are obtained, and the bifurcation surfaces (including codimension-1 homoclinic bifurcation surfaces, double periodic orbit bifurcation surfaces, homoclinic-doubling bifurcation surfaces, period-doubling bifurcation surfaces and codimension-2 triple periodic orbit bifurcation surface, and homoclinic and double periodic orbit bifurcation surface) and the existence regions are also located.

The local moving frame approach is employed to study the bifurcation of a degenerate heterodimensional cycle with orbit-flip in its nontransversal orbit. Under some generic hypotheses, we provide the conditions for the existence, uniqueness and noncoexistence of the homoclinic orbit, heteroclinic orbit and periodic orbit. And we also present the coexistence conditions for the homoclinic orbit and the periodic orbit. But it is impossible for the coexistence of the periodic orbit and the persistent heterodimensional cycle or the coexistence of the homoclinic loop and the persistent heterodimensional cycle. Moreover, the double and triple periodic orbit bifurcation surfaces are established as well. Based on the bifurcation analysis, the bifurcation surfaces and the existence regions are located. An example of application is also given to demonstrate our main results.

Bifurcations of homoclinic orbit with orbit-flip and resonant eigenvalues corresponding to the tangent directions are investigated in a four-dimensional system. The existence, number, coexistence and incoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are given, and the bifurcation surfaces and the existence regions are also located.

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The multiplicity of solutions for a (p, q)-Laplacian equation involving critical exponent p uq u = λV(x)|u| k-2 u + K(x)|u| p *-2 u, x ∈ R N is considered. By variational methods and the concentration-compactness principle, we prove that the problem possesses infinitely many weak solutions with negative energy for λ ∈ (0, λ *). Moreover, the existence of infinitely many solutions with positive energy is also given for all λ > 0 under suitable conditions.