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NONTRIVIAL SOLUTIONS OF THE DIRICHLET PROBLEM FOR A CLASS OF NONLINEAR ELLIPTIC EQUATIONS
Chen Zuchi, Shen Yaotian
Acta mathematica scientia,Series B. 1987, 7 (1):
63-74.
Under the mild conditions by using the theory of the generalized Sobolev space and minimax theory we get the existence of solutions for a wide class of nonlinear Dirichlet problem as follows d/dxi[Gi(x,Du)]+f(x,u)=0, x∈Q, u|∂Q=0, where Q is a bounded domain in Rn,Gi(x,q)=∂G/∂qi, q=(q1,q2,…,qn). This equation is the Euler equation of the functional I(u)=∫Q[G(x,Du)-F(x,u)]dx, where F(x,s)=∫0xf(x,t)dt.
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