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This thesis is devoted to prove the existance of solution and structure of solu­ tion for some partial differential equations by using some modern topological and variational thechniques. Taking direction from the literature, this thesis is interested in existence, uniqueness, blow-up in finite time for some evolution equations, multiplicity and radial solution for certain elliptic partial differential equations.-Employing Fibrering, Galerkins, Mountain Pass-Lemma and lions com­ pactness Lemma are sharp in this thesis to prove the exsistence and multiplicity of solutions and overcome lack of compactness in some cases  

We discuss, in the setting of UMD Banach spaces E, existence, uniqueness and the continuous dependence on the data A, F, G and of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form dX(t) = [AX(t) + F(t;X(t))]dt + G(t;X(t))dWH(t); t 2 [0; T]; X(0) = ; Where A generates an analytic C0-semigroup on a UMD Banach space E and WH is a cylindrical Brownian motion with values in a Hilbert space H. We can see that if the mappings F : [0; T] E ! E and G : [0; T] E ! L(H;E) satisfy suitable Lipschitz conditions and is F0-measurable and bounded, then this problem has a unique mild solution.And We arw going to review continuous dependence of the solutions...