By applying the method of monotone operators and techniques related to the method of sub-super solutions, we conduct a thorough examination of the existence and uniqueness of solutions to mixed local-nonlo cal Dirichlet boundary value problems. These problems are characterized by the presence of both local and nonlo cal operators, which introduce significant complexity into the analysis. In addition, we also investigate the continuous dependence of these solutions on various parameters. To achieve this, we derive a parameter-dependent surjectivity theorem in presence of non-uniqueness of solutions. Furthermore, we extend our analysis to include applications where the problem depends on a bounded weight.
We are concerned with the following gauged nonlinear Schrodinger equations {-triangle u + (1 + |x|(2))u + lambda (integral(infinity )(|x|)h(u)(s)/su(2 )(s)ds + h(u)(2)(|x|)/|x|(2))u = g(x, u) in R-2, u(x) = u(|x|), where lambda > 0 is a parameter, h(u)(s) = integral(s )(0)r/2u(2)(r)dr and the nonlinearity g has the supercritical exponential growth at infinity in the Trudinger-Moser sense. When the continuous function g(x, & sdot;) is odd for every x is an element of R-2, thanks to a new type of critical point theorem, we conclude the existence of multiple nontrivial solutions for a sufficiently small lambda > 0 with the help of a powerful truncation procedure and the elliptic regularity theory. In particular, the analyses developed in this article allow us to contemplate the case that g(x, u) = |u|(p-2)u with 2 < p <= 6. Moreover, we also demonstrate that the "almost optimal" growth condition lim(|t|->+infinity) G(x, t)e(-4xt2 )> 0 uniformly in x is an element of R-2 makes the existence of a mountain-pass solution available, where and in the whole paper G(x, & sdot;) is denoted by the primitive of g(x, & sdot;).
In this paper, we focus on prescribed mass standing waves for the nonlinear Schrodinger equation with rotation and it is natural to consider the normalized solutions for the stationary equation with rotation -Delta u +V(x)u-Omega Lu = & micro;qu +a|u|qu in R-2, having prescribed mass integral(2)(R) |u|(2)dx = 1, where a > 0, q > 2, & micro;(q) is an element of R, V(x) = |x|(2) is a trapping potential, Omega is an element of (0,2) characterizes the rotational velocity of Bose gas rotates around the origin, and L := -i(x1 partial derivative x(2)-x(2)partial derivative x(1)) is the angular momentum. It is shown that there exist two solutions for q > 2 approaching to the mass critical exponent 2, one of which is a local minimizer and the other is mountain pass type. In particular, we prove that the local minimizer u(q) converges to the ground state of above problem with q = 2 as q -> 2(+). Moreover, we verify that the mountain pass type solution occurs blow-up as q -> 2(+.)
We study the following type of magnetic system-(del + iA(x))2u + V(x)u + lambda phi|u|(x)u = |u|p-1u, x is an element of I3, (0.1) where p is an element of (1, 2), lambda is a parameter, and phi|u|(x) is a nonlo cal convolution potential. Under suitable assumptions on the potentials A(x) and V (x), we show that problem (0.1) has a ground state by using variational methods. Moreover, the asymptotical behavior of ground states as lambda -> 0 has also been discussed.
In this paper, we study the existence of bound states with prescribed mass for the NLS energy with localized nonlinearities which combine L2-subcritical and supercritical terms on a noncompact metric graph G. The study employs a min-max principle for constrained functionals, integrating the monotonicity technique with second-order details of Palais-Smale sequences, along with an analysis of blow-up in bound states characterized by a given mass and a uniformly bounded Morse index. Our results extend the previous results of J. Borthwick, X. Chang, L. Jeanjean, and N. Soave to nonlinear terms with mixed L2-subcritical and supercritical growth.
