Abstract:
The present work in the thesis deals with the existence and qualitative questions to biharmonic boundary value problems. More specifically, we establish the existence of solution to several type of biharmonic equations and systems. We also generalise several celebrated identities and inequalities to the biharmonic operator setting, such as Picone�s identity, Caccioppoli�s, Adams� inequalities and also consider some applications. This work is motivated by the interest of many researchers in the study of biharmonic equations. The Biharmonic equation D2u = f with suitable boundary data models various physical problems related to plates, e.g., Dirichlet boundary conditions (u=0= �u �n on boundary) model clamped plates, Navier boundary conditions (u = 0 = Du on boundary) model hinged plates etc. First, we consider the problem of the existence of a positive solution to the following biharmonic system: where B denotes the unit ball in Rn with boundary �B, l is a positive parameter, a;b : W ! R are sign changing potentials, f ;g : [0;�) ! R are continuous with f (0) > 0;g(0) > 0. We prove the existence of a positive solution to (0.1) with the aid of the Leray-Schauder fixed point theorem, by utilising the representation of the solution to (0.1) in a ball. We also establish the existence of a unique weak solution to the following system of biharmonic equations: without monotonicity assumptions on the nonlinearities, in a smooth and bounded domain W: We assume that a;b 2 L�(W) are sign changing potentials. We use theory of monotone operators to prove our existence results. Further, we establish the existence of a unique weak solution to the following singular biharmonic system: where 0 2 W Rn; n 5 and a;b 2 L�(W) are sign changing potentials. We prove our result by utilising the theory of monotone operators and the Rellich inequality. Next, we consider the problem of the existence of a solution to the following biharmonic equation:where W R2 is a smooth and bounded domain, a 0; l > 0; m > 0; n > 0; a;b; f 2 L1(W); r 1 and 0 < q < 1: The special feature of this problem is that it involves the determinant of the Hessian with sign changing concave, quadratic and gradient nonlinearities. We remark that nonlinearities of the type ma(x)up+nb(x)uq; where 0<q<1< p<� are called concave-convex nonlinearities. The main difficulties arising in these problems are due to the determinant of the Hessian of the function and the involvement of sign changing concave, quadratic and gradient nonlinearities. Because of the presence of the gradient term, the problem is not variational, in general. To overcome this difficulty, we make use of an iteration technique introduced by Figueiredo et.al. We employ certain inequalities allowing to place the determinant of the Hessian in Hardy space and prove the existence and multiplicity of solutions by using critical point theory. We also deal with Picone�s identity for the biharmonic operator on the Heisenberg group. Picone�s identity plays an important role in the qualitative theory of elliptic PDEs. The classical Picone�s identity says that if u and v are differentiable functions such that v > 0 and u 0; then (0.2) has numerous applications to second-order elliptic equations and systems. We extend the identity (0.2) to biharmonic operators and also establish its nonlinear analogue. As an application of it, we establish a Hardy type inequality, a Picone inequality, a Caccioppoli inequality and the monotonicity of the first eigenvalue of the associated problem. We also obtain Picone�s identity for operators of form ??div(a(x;u;�u)); whose coefficients a : W R Rn !Rn satisfy certain hypotheses. This operator generalises the Laplace and p-Laplace operators. As an application of Picone�s identity, we establish a Hardy type inequality, a Sturmian comparison theorem, the monotonicity property of the first eigenvalue, nonexistence of positive supersolutions and a Caccioppoli inequality. Further, we establish an Adams type inequality for the biharmonic operator on the Heisenberg group. Let W Rn; n 4 be a bounded domain. The Sobolev embedding theorem says that for p < n;W2;p 0 (W) ,! Lq(W); 1 q np n??p : For the limiting case p = n; we haveW2;n 0 ,!Lq(W); 1 q < � but it is well known thatW2;n 0 (W) 6,!L�(W): Then there is a natural question that what is the smallest possible space in which, we have embedding of W2;n 0 (W)? This question was answered by D.R. Adams in the case of bounded domains in Rn: We extend the Adams� inequality to the case of bounded domains in Heisenberg group H: We make use of convolution results involving Reisz potential established by W.S. Cohn and G. Lu. We also establish an Adams type inequality with singular potential. As an application of Adams� inequality, we prove the existence of solution to the following biharmonic equation with Dirichlet�s boundary condition on the Heisenberg group: where 0 2 W H is a bounded domain, 0 a Q; Q = 4 is the homogeneous dimension of H and the nonlinearity f satisfies certain hypotheses. The problem (0.3) has the following special features, which make it challenging to study. First, it contains the nonlinearity f ; which is of exponential growth and the potential 1 r(x )a ; 0 a Q; which has singularity at r(x ) = 0: This problem is handled with the use of Adams� inequality with singular potential. Second, the case a = Q; is critical for the potential. Since we do not have a singular Adams type inequality in the case a = Q; we use an approximation method. More precisely, we approximate (0.3) with a sequence of problems which are subcritical in the potential, i.e., a < Q and then, we pass to the limit to conclude that (0.3) has a nontrivial solution in the case a = Q: Finally, we establish the semi-stability of positive solutions to the following problem: where W Rn is a smooth and bounded subset. The nonlinearities of the type a(x)u?? f (x;u) are known as logistic nonlinearities. There have been several research works to study problems related to the existence and stability of solutions to the Laplace equation with logistic nonlinearities. To show the semi-stability of u; we show that the eigenvalues of the linearised operator associated with (0.4) at the point u are non-negative.