Elliptic and parabolic semilinear problems without conditions at infinity
Let \(m\geq 1\) and \(N\geq 1\) be integers and let \(q=(p+1)/p\) be the conjugate Hölder exponent of a number \(p+1>2\). For \(f\in L^ q_{loc}({\mathbb{R}}^ N)\), one principal result is the existence of a distribution \(u\in W^{m,2}_{loc}({\mathbb{R}}^ N)\cap L^{p+1}_{loc}({\mathbb{R}}^ N),\) satisfying \((-\Delta)^ mu+| u|^{p-1}u=f\) in \({\mathbb{R}}^ N\). This solution u is unique if, in addition, \(p\leq (N+2m)/(N-2m).\) Similar theorems are proved for the parabolic equation \[ u_ t+(-\Delta)^ mu+| u|^{p-1}u=f. \] Such problems require neither conditions at infinity nor limitations on the growth of f at infinity. The proofs are based on local estimates obtained from a new weighted Gårding-type inequality (related to a Caccioppoli-type inequality) not requiring boundary conditions. Additional theorems concern elliptic equations with strong nonlinearities of order zero and equations with unbounded measures f. Second order results of the present type were obtained by \textit{H. Brezis} [Appl. Math. Optimization 12, 271-282 (1984; Zbl 0562.35035)].
scientific article; zbMATH DE number 4111167
scientific article