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Dual ground state solutions for the critical nonlinear Helmholtz equation. (English) Zbl 1444.35036

The authors consider the following problem involving the critical Helmholtz equation \[ -\Delta u-k^2 u=Q(x)|u|^{2^*-2}u, \quad u\in W^{2,2^*}(\mathbb{R}^N), \tag{1} \] where \(Q:\mathbb{R}^N\rightarrow \mathbb{R}\) is a nonzero nonnegative bounded function, and \(2^*:=2N/N-2\) is the critical Sobolev exponent. For \(N\geq 4\), the authors prove that this problem admits a (strong) nontrivial real valued solution provided that the weight function \(Q\) satisfies the following additional properties
\(Q=Q_{\mathrm{per}}+Q_0\), with \(Q_{\mathrm{per}},Q_0\) nonnegative, \(Q_{\mathrm{per}}\) periodic, and \(\displaystyle{\lim_{|x|\rightarrow +\infty}Q_0(x)=0}\);
there exists a global maximum point \(x_0\) of \(Q\) such that, as \(x\rightarrow x_0\),
\[ Q(x_0)-Q(x)=o(|x-x_0|^2),\text{ if }N\geq 5, \] and \[ Q(x_0)-Q(x)=O(|x-x_0|^2),\text{ if }N=4. \] By classical results, it is known that the solutions of the critical Helmholtz equation are not in \(W^{1,2}(\mathbb{R}^N)\) in general, so that no solution in \(\mathbb{R}^N\) can be found as critical point of the associated natural energy functional.
Actually, the authors prove the existence of a solution in \(W^{2,2^*}(\mathbb{R}^N)\) by considering the dual energy functional \[ J_Q(v)=\frac{N+2}{2N}\int_{\mathbb{R}^N}|v|^{\frac{2N}{N+2}}dx-\frac{1}{2}\int_{\mathbb{R}^N}Q^{\frac{1}{2^*}}v\mathbf{R}_k(Q^{\frac{1}{2^*}}v)dx, \ \ \ v\in L^{\frac{2N}{N+2}}(\mathbb{R}^N), \] where \(\mathbf{R}_k\) denotes the real part of the resolvent operator of \(-\Delta -k^2\), whose critical points \(v\) are related to the solutions \(u\) of problem (1) via the transformation \[ u=\mathbf{R}_k(Q^{\frac{1}{2^*}}v). \] The functional \(J_Q\) has a mountain pass geometry and the authors are able to find a critical point of \(J_Q\) via a mountain pass theorem after showing that \(J_Q\) satisfies the Palais-Smale condition at each level \(\beta \in ]0,L_Q^*[\), where \(L_Q^*\) is the least energy level of the limiting problem \[ -\Delta u=Q(x_0)|u|^{2^*-2}u,\text{ in }\mathbb{R}^N, \] on varying of \(x_0\in \mathbb{R}^N\), and that the mountain pass level \(L_Q\) of \(J_Q\) satisfies \(L_Q\)<\(L_Q^*\). The proof of this latter inequality needs some accurate estimates involving Bessel functions.
Since the found critical point \(v\) of \(J_Q\) has minimal energy among all nontrivial critical points, the corresponding solution \(u\) of \((1)\) is called dual ground state solution. In contrast to the case \(N\geq 4\), the authors show that, for \(N=3\), no dual ground state solution of problem \((1)\) can exist.

MSC:

35J20 Variational methods for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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