Two sequences of solutions for the semilinear elliptic equations with logarithmic nonlinearities

摘要

We are interested in the following elliptic equation @@@ {-Delta u = a(x)u log vertical bar u vertical bar, x is an element of Omega, (0.1) @@@ u = 0, on partial derivative Omega, @@@ where Omega is a bounded domain of R-N (N >= 2) with smooth boundary partial derivative Omega, and a(x) is an element of C(Omega). The existence and multiplicity of solutions are obtained by using variational methods. Quite surprisingly, the existence of solutions is deeply influenced by the sign of a(x). More precisely, @@@ (i) if a(x) > 0, equation (0.1) possesses a sequence of solutions whose energy and H-0(1)(Omega)-norms diverge to positive infinity; @@@ (ii) if a(x) < 0, equation (0.1) possesses a sequence of solutions whose energy and H-0(1)(Omega)-norms converge to zero; @@@ (iii) if a(x) is sign-changing, equation (0.1) possesses two sequences of solutions: one sequence of solutions is with energy and H-0(1)(Omega)-norms diverging to positive infinity, while the other one is with energy and H-0(1)(Omega)-norms converging to zero.

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