The Existence And Multiplicity Of Solutions For Several Classes Of Elliptic Equations With Jumping Nonlinearities

Jumping nonlinearities problem roots in the study on light-wave and electromagneticwave in physics. It reflects the resonance and oscillation phenomena and has practicalapplications in physics and economics fields. So, researchers show increasing interest inthe problem and have achieved many results. However, most of jumping nonlinearitiesproblems are limited to the study on the relationship between the asymptotic limit ofnonlinearities and the eigenvalues. The relationship between the the asymptotic limitand Fucík spectrum can reflect more actual phenomena. Therefore, the study on thesolutions when the nonlinearity of partial differential equation oscillates with respect toFucík spectrum has the important practical significance.In this paper, based on the theory of Fucík spectrum, we study the following fourjumping nonlinearities problems relying on the fundamental theory of nonlinear func-tional analysis with a combination of sub-super solutions methods, critical point theory,degree theory, mountain pass theorem in order intervals by using the nonlinearities ofpartial differential equations oscillating with respect to Fucík spectrum:1. We discuss the solutions for a class of Laplace equations with jumping nonlinear-ities and Neumann boundary condition. We consider the case that the asymptotic limitsof the nonlinearity fall in the Type I_l（l>2） and II_l（l≥1） regions formed by the curvesof the Fucík spectrum of Laplace operator. Firstly, we choose some order intervals whichhave two pairs of strict constant sub-super solutions. We obtain at least a mountain passcritical point based on the the truncation technique and critical point theory. It is difficultto prove whether the critical point is constant. We overcome the difficulty by using thecritical groups of Fucík spectrum. Next, we obtain the existence of at least two noncon-stant solutions in the order intervals by computing the Morse index combined with degreetheory. Then, we get the multiplicity of sign-changing solutions using degree theory andthe method of sub-supersolution; also we obtain a sequence of critical points, which areassociated with a sequence of negative energies going toward positive infinity.2. We discuss the solutions for a class of Laplace equations with jumping nonlin-earities and Robin boundary condition. Firstly, based on the existence of a positive su-persolution and a negative subsolution, using mountain pass theorem in order intervals and critical point theory, we obtain a nontrivial solution. Combining with degree the-ory, we get the existence of at least four nontrivial solutions. Because of the limits ofRobin boundary, we overcome the difficulty of constructing the order intervals by us-ing mountain pass theorem in half intervals （the supersolution case）. Finally, we obtainthe multiplicity of corresponding resonance equations by mountain pass theorem in halfintervals.3. We discuss the solutions for a class of quasilinear equations with jumping non-linearities and Dirichlet boundary condition. Note that p-Laplace operator has more com-plicated nonlinearities than Laplace operator and we know little about the spectrum andproperty of p-Laplace operator in W₀（1-p）（Ω） until now. We overcome the limits of operatorspectrum by using the eigenvalues according to Z₂-cohomological index. We study thegeneral resonance case in the sense that λ_k<a, b <λ_k+1 for two consecutive variationaleigenvalues. We obtain at least a nontrivial solutions by constructing cohomological localsplitting regions based on cohomological local splitting theory and variational arguments.As for the abstract setting, we establish the existence of at least a nontrivial solutions byusing the similar methods.4. We discuss the solutions for a class of p-Laplace equations with jumping non-linearities. Based on the Fucík spectrum theory of p-Laplace operator, we obtain themultiplicity of nontrivial solutions and sign-changing solutions with Neumann and Robinboundary condition respectively by using the similar methods above. It is a relatively newconclusion relative to the study on the solutions of p-Laplace with jumping nonlinearities.