Existence And Multiplicity Of Solutions For Heat Equation And Klein-Gordon-Maxwell System

In this paper, we use variational method and analysis technologies to study two kinds of problem which have strong physical background (heat equation and Klein-Gordon-Maxwell system). We will find the existence and multiplicity of solutions for these kind of problems. More precisely, we consider the heat equation in Chapter 2 as follows: where A is a parameter. b(x) is a sign-changing function in RN and 1< q< 2*, where 2* is the critical Sobolev exponent. We consider the existence of nonnegative solutions to this type problem involving a sign-changing weight. At the same time, some bifurcation results and non-existence results are also established. As it is well known, when one uses the variational methods to find the critical points of the functional, some geometry structures are needed, for example, the Mountain Pass structure, the Linking structure and etc. As for our problem, the difficulty lies in that the functional may not possess such structures because of the sign-changing weight. In order to overcome this difficulty, we turn to another approach, i.e., the Nehari manifold. We first study the superlinear case, that is 2< q< 2*, next, we consider the sublinear case, i.e., the case 1< q< 2. At last we get one, two and three solutions for this equation. Later, in Chapter 3, we consider the following equation where 2* is the critical Sobolev exponent,2< p< q< 2* and a or b is a sign-changing function. Under different assumptions on a and b we prove the existence of infinitely many solutions to above problem. We also show that one of these solutions is non-negative. Under some assumptions, we show that the energy functional satisfies the Palais-Smale condition on the Nehari manifold and then using some arguments based on the Krasnoselskii genus, we establish the existence of infinitely many solutions. In Chapter 4, we consider the equation where f∈C(RN xR,E), N≥3. We apply the variational methods to obtain the existence of ground state solutions when nonlinearity is superlinear and asymp-totically linear at infinity, respectively. In Chapter 5, we consider the following Klein-Gordon-Maxwell system where w> 0 is a constant, u,φ:R3â†'R, V:R3â†'R is a potential function. Using critical point theory, we establish sufficient conditions for the existence of infinitely many solitary waves solutions. Results obtained complement and improve the existing ones X.-M. He [Multiplicity of Solutions for a Nonlinear Klein-Gordon-Maxwell System. Acta Appl. Math..130:237-250,2014]. Moreover, problem can be compared with recent interesting studies on the Schrodinger-Poisson equation with non-constant potential and generalizes these results. In the last chaper, we consider a class of resonance elliptic system and obtain infinitely many nontrivial solutions when the nonlinearity with sublinear growth and the proof is very simple.