Further results on quasiperiodic partially localized solutions of homogeneous elliptic equations on R^N+1

We study positive partially localized solutions of the elliptic equation Δ_xu+u_yy+f(u)=0,(x,y)∈R^N×R, where N≥2 and f is a C¹ function satisfying f(0)=0 and f^′(0)<0. By partially localized solutions we mean solutions u(x,y) which decay to zero as |x|→∞ uniformly in y. Our main concern is the existence of positive partially localized solutions which are quasiperiodic in y. The fact that such solutions can exist in equations of the above form was demonstrated in our earlier work: we proved that the nonlinearity f can be designed in such a way that equation (1) possesses positive partially localized quasiperiodic solutions with 2 frequencies. Our main contributions in the present paper are twofold. First, we improve the previous result by showing that positive partially localized quasiperiodic solutions with any prescribed number n≥2 of frequencies exist for some nonlinearities f. Second, we give a tangible sufficient condition on f which guarantees that equation (1) has such quasiperiodic solutions, possibly after f is perturbed slightly. The condition, with n=2, applies, for example, to some combined-powers nonlinearities f(u)=u^p+λu^q−u with suitable exponents p>q>1 and coefficient λ>0.